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CausalBind: Causal Modeling and Learning for Protein-Molecule Virtual Screening
Loka Li, Jin Tian, Kun Zhang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 10/7/2026, 9:10:30 AM
Summary
The paper introduces CausalBind, a framework for protein-molecule virtual screening that leverages causal modeling to identify sparse interaction patterns rather than relying on dense holistic alignment. By formalizing binding as a V-structure causal model under Heckman-style selection, the authors prove that sparse interactions are identifiable under specific structural constraints. CausalBind implements these principles through three variants (SP, LR, EMB) and demonstrates superior performance and generalization on DUD-E and LIT-PCBA benchmarks compared to existing retrieval-based methods.
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Relation Signals (9)
CausalBind → evaluatedon → LIT-PCBA
confidence 95% · Extensive experiments on DUD-E and LIT-PCBA benchmarks
CausalBind → evaluatedon → DUD-E
confidence 95% · Extensive experiments on DUD-E and LIT-PCBA benchmarks
CausalBind → includesvariant → CausalBindEMB
confidence 95% · CausalBindEMB applies the constrained mask after pooling
CausalBind → includesvariant → CausalBindSP
confidence 95% · we propose CausalBind with three implementation variants... the main CausalBindSP
CausalBind → includesvariant → CausalBindLR
confidence 95% · CausalBindLR uses a low-rank factorization
CausalBind → basedontheory → V-structure causal model
confidence 90% · formalize this prior via a V-structure causal model under Heckman-style selection
CausalBind → outperforms → Retrieval Baselines
confidence 90% · all variants consistently outperform strong retrieval baselines
CausalBindSP → usesmechanism → sparse concept-pair mask
confidence 90% · CausalBindSP uses a concept extractor and a learnable sparse concept-pair mask
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Abstract
Abstract:Protein-molecule virtual screening is increasingly cast as a problem of representation learning in a shared embedding space. Existing methods rely on dense holistic alignment, entangling invariant binding determinants with nuisance correlations and limiting transfer to new targets. It has been noted that binding in protein-molecule systems involves sparse cross-modality interactions: binding is governed by a small contact interface and a few decisive local interactions (e.g., hydrogen bonds, hydrophobic contacts, and salt bridges) rather than the global structures of the protein and molecule. We hypothesize that uncovering and leveraging sparse interaction patterns is critical for generalization beyond the training data, as these patterns are reusable and expected to improve performance across different scenarios. In this paper, we aim to identify and leverage sparse interaction patterns, and verify our hypothesis. Since the training data contain only observed binding pairs, we formalize this prior via a V-structure causal model under Heckman-style selection, and establish three theoretical results: (i) the latent concepts of interacting proteins and molecules are not identifiable without appropriate sparsity constraints; (ii) these concepts and their sparse interactions are component-wise identifiable under structural sparsity conditions; and (iii) a low-rank relaxation of these conditions yields subspace identifiability of the concepts and interactions. Inspired by these principles, we propose CausalBind with three implementation variants. Extensive experiments on DUD-E and LIT-PCBA benchmarks show that all variants consistently outperform strong retrieval baselines, with the largest gains on LIT-PCBA early enrichment, and further generalize to target- and scaffold-level out-of-distribution splits. Code is available at this https URL.
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- Source: https://arxiv.org/abs/2610.07340v1
- Canonical: https://arxiv.org/abs/2610.07340v1
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_set:Ne tcboxmath _set:Ne CausalBind: Causal Modeling and Learning for Protein-Molecule Virtual Screening Loka Li Affiliation: Mohamed bin Zayed University of Artificial Intelligence Jin Tian Affiliation: Mohamed bin Zayed University of Artificial Intelligence Kun Zhang Affiliation: Mohamed bin Zayed University of Artificial Intelligence Affiliation: Carnegie Mellon University Abstract Protein-molecule virtual screening is increasingly cast as a problem of representation learning in a shared embedding space. Existing methods rely on dense holistic alignment, entangling invariant binding determinants with nuisance correlations and limiting transfer to new targets. It has been noted that binding in protein–molecule systems involves sparse cross-modality interactions: binding is governed by a small contact interface and a few decisive local interactions (e.g., hydrogen bonds, hydrophobic contacts, and salt bridges) rather than the global structures of the protein and molecule. We hypothesize that uncovering and leveraging sparse interaction patterns is critical for generalization beyond the training data, as these patterns are reusable and expected to improve performance across different scenarios. In this paper, we aim to identify and leverage sparse interaction patterns, and verify our hypothesis. Since the training data contain only observed binding pairs, we formalize this prior via a V-structure causal model under Heckman-style selection, and establish three theoretical results: (i) the latent concepts of interacting proteins and molecules are not identifiable without appropriate sparsity constraints; (i) these concepts and their sparse interactions are component-wise identifiable under structural sparsity conditions; and (i) a low-rank relaxation of these conditions yields subspace identifiability of the concepts and interactions. Inspired by these principles, we propose CausalBind with three implementation variants. Extensive experiments on DUD-E and LIT-PCBA benchmarks show that all variants consistently outperform strong retrieval baselines, with the largest gains on LIT-PCBA early enrichment, and further generalize to target- and scaffold-level out-of-distribution splits. Code and scripts to reproduce our experiments are publicly available from GitHub at https://github.com/lokali/CausalBind. 1 Introduction Modern drug discovery faces a fundamental challenge: identifying promising drug candidates from libraries containing millions to billions of candidate compounds (Berdigaliyev and Aljofan, 2020). Virtual screening (Walters and Wang, 2020), which prioritizes protein-molecule pairs before costly experimental validation, has become a major bottleneck in this pipeline and a central problem in AI for drug discovery (Jia et al., 2026). An overview of the broader pipeline is illustrated in App. A1. Figure 1: Dense alignment versus sparse binding. (a) Holistic dense alignment. Existing retrieval-based virtual screening methods compare every protein surface feature with every molecule feature in the shared embedding space; this “full-surface” alignment entangles binding-relevant signals with nuisance correlations. (b) Biology-inspired sparse binding. Protein-molecule binding is governed by a small contact interface and a handful of key local interactions (e.g., hydrogen bonds, hydrophobic contacts, salt bridges, and π-stacking), which together account for the majority of the binding affinity. Existing virtual-screening methods broadly fall into three regimes: physics-based docking systems (Friesner et al., 2004; Trott and Olson, 2010) that score complexes via force fields; structure-aware machine-learning scoring models (McNutt et al., 2021; Shen et al., 2022; Shen et al., 2023; Cao et al., 2024; Moon et al., 2022; Lu et al., 2022; Brocidiacono et al., 2023) that learn to rank protein-molecule complexes from data; and retrieval-based representation learners (Gao et al., 2023; Han et al., 2025; Feng et al., 2025; Wang et al., 2026; He et al., 2026) that embed both modalities into a shared space. The first two reason explicitly about three-dimensional poses and are valuable for mechanistic analysis, but they are typically better suited to rescoring a small candidate set than to ultra-large-scale retrieval. Retrieval-based methods, by contrast, embed both molecules and proteins into a shared embedding space and screen by similarity, scaling naturally to libraries with millions of compounds and now consistently dominating the empirical leaderboard. See App. A2 for extended related work. Despite this success, retrieval models still learn holistic representations: they align full molecule and protein embeddings without separating binding-relevant determinants from nuisance factors. This reflects a structural mismatch with the biological process, as illustrated in Figure 1, where the binding decision is governed by a small contact interface and a few decisive local interactions rather than the global structures of the protein and molecule. A classical example is the notion of “hot spots”, where a small subset of interfacial contacts contributes the majority of binding free energy (Clackson and Wells, 1995; Bogan and Thorn, 1998). As a result, holistic models are prone to capturing spurious global correlations rather than the local structural features that actually drive binding (Chen et al., 2019; Graber et al., 2025), which may hinder generalization to novel protein families and unseen chemical scaffolds. Additional details about biological observations are provided in App. A3. These limitations motivate the core question to be answered in this paper: if protein-molecule binding is driven by a small number of local compatibilities, should one represent virtual-screening in terms of the sparse concept-level interactions, instead of potentially dense holistic alignment, and leverage this representation for better generalization ability? Our results strongly suggest that the answer is yes. To analyze this issue, we first develop a V-structure causal model of binding under Heckman-style selection (Schölkopf et al., 2021; Zhang et al., 2026), and establish three identifiability results for the sparse protein-molecule interaction: dense cross-modal interaction is generally non-identifiable, sparse antichain interaction is component-wise identifiable up to trivial equivalences, and under a low-rank condition the interaction modes are subspace-identifiable. Building on this theory, we propose CausalBind as a flexible structure-constrained representation framework with three different implementations: the main CausalBindSP uses a concept extractor and a learnable sparse concept-pair mask, CausalBindLR uses a low-rank factorization of the same concept-axis mask, and CausalBindEMB applies the constrained mask after pooling in embedding space. Our main technical contributions are threefold. (i) Theoretically, we provide an identifiability analysis of representation learning for protein-molecule binding under binder-only selection, showing that dense interactions are not identifiable, whereas sparse antichain-structured interactions are component-wise identifiable, and low-rank interactions are subspace-identifiable. (i) Methodologically, we instantiate these principles through three CausalBind implementations within a concept-based retrieval framework. CausalBindSP routes binding evidence through a learnable sparse concept-pair mask, CausalBindLR imposes an explicit low-rank concept-axis factorization, and CausalBindEMB applies the constraint in pooled embedding space. (i) Empirically, on DUD-E and LIT-PCBA benchmarks, all three CausalBind implementations outperform reproduced baseline methods, with the largest relative gains on the property-debiased LIT-PCBA early enrichment. Additional analyses, including multi-seed ablations, hyperparameter studies, antichain diagnostics of the learned masks, out-of-distribution evaluation on DEKOIS 2.0, an interpretability case study, and affinity-ranking results on additional benchmarks, support the efficacy of our method. 2 Theoretical Analysis: Sparse Binding under Heckman Selection This section develops a causal perspective on protein-molecule binding under binder-only observation. In § @sectionsign 2.1, we model binding as a V-structure and study identifiability (Daunhawer et al., 2023) under Heckman-style selection (Heckman, 1979). Our theorems in § @sectionsign 2.2 proceed in three phases, from non-identifiability to component-wise and subspace identifiability. 2.1 Causal Model: Binding as a Selected Collider cm1c_m^1cm2c_m^2cm3c_m^3cp1c_p^1cp2c_p^2cp3c_p^3111100B=1B=1 Figure 2: Binder-selected sparse-interaction view. Observing only binders (B=1B=1) induces cross-modal dependence through the collider B. Green edges denote active local interactions. Red dashed edges blocked interactions. Purple dashed curves allowed within-modality dependence. The causal model combines the biological sparsity motivation above with the binder-only nature of training data. In practice, we predominantly observe pairs with B=1B=1; conditioning on this collider can make molecule and protein features statistically dependent even when they are marginally independent. Figure 2 summarizes this collider structure of sparse interactions. Let xM∈Mx^M ^M and xP∈Px^P ^P denote observed molecule and protein, generated from latent concepts xM=gM(M,εM),xP=gP(P,εP), [rgb]0,0,0x^M=g^M(c^M, ^M), x^P=g^P(c^P, ^P), (1) where M=(c1M,…,cKMM)c^M=(c^M_1,…,c^M_K_M) and P=(c1P,…,cKPP)c^P=(c^P_1,…,c^P_K_P) are modality-specific atomic features, and εM,εP ^M, ^P denote modality-specific observation noise or nuisance variation. Here KMK_M and KPK_P count the number of concepts, respectively. Binding arises through cross-modal interactions. Motivated by existing literature (Wang et al., 2026; Feng et al., 2025), we model binding as a Bernoulli collider: B∼Bernoulli(σ(∑i=1KM∑j=1KPℳijϕ(ciM,cjP))),xM←M→B←P→xP, [rgb]0,0,0B \! (σ\! ( _i=1^K_M _j=1^K_PM_ij\,φ(c^M_i,c^P_j) ) ), [rgb]0,0,0x^M ^M→ B ^P→ x^P, (2) where ℳ∈ℝ≥0KM×KPM _≥ 0^K_M× K_P is a (typically sparse) interaction mask; its scale is identifiable only jointly with ϕφ. In the theory, each atomic concept is a scalar latent variable, i.e., ciM,cjP∈ℝc_i^M,c_j^P , and ϕ:ℝ×ℝ→ℝφ:R×R maps one molecule concept and one protein concept to a scalar logit-level compatibility score; σ(⋅)σ(·) is the logistic sigmoid. The nuisance variables affect only the observed modalities and are marginalized in p(xM,xP∣B=1)p(x^M,x^P B=1); the binding variable depends on the recovered concepts. Before selection, M⟂Pc^M \!\!\! ^P. After conditioning on B=1B=1, the two modalities become dependent. Here we denote the distribution of the observed training data as: p(xM,xP)=p(xM,xP∣B=1)p^D(x^M,x^P)\;=\;p(x^M,x^P B=1), which generally induces dependence between xMx^M and xPx^P. Furthermore, by Bayes’ rule, we have p(M,P)∝p(M)p(P)σ(∑i=1KM∑j=1KPℳijϕ(ciM,cjP)). [rgb]0,0,0p^D(c^M,c^P)\; \;p(c^M)\,p(c^P)\,σ\! ( _i=1^K_M _j=1^K_PM_ijφ(c^M_i,c^P_j) ). (3) This induced dependence is the only signal available for recovering (ℳ,ϕ)(M,φ). Below we give the definition of identifiability used in this paper, followed by some common assumptions. Definition 1 (Identifiability). Identifiability is concerned with whether the latent concepts and interaction structure are uniquely determined by the observed binder-selected distribution p(xM,xP)p^D(x^M,x^P), modulo the unavoidable equivalences of the chosen representation. In the theory, the concept dimensions KMK_M and KPK_P are treated as given model dimensions. We use two levels of identifiability. 1. Component-wise identifiability. The concepts and sparse interaction support are component-wise identifiable if any two parameterizations (gM,gP,ℳ,ϕ)(g^M,g^P,M,φ) and (g~M,g~P,ℳ~,ϕ~)( g^M, g^P, M, φ) inducing the same p(xM,xP)p^D(x^M,x^P) differ only by separate molecule-side and protein-side concept-index permutations, component-wise invertible transformations of individual concepts, and the trivial joint rescaling (ℳ,ϕ)↦(aℳ,a−1ϕ)(M,φ) (aM,a^-1φ) for a>0a>0. 2. Subspace identifiability. Suppose the interaction kernel has a minimal rank-r factorization S(M,P)=(M)⊤(P)S(c^M,c^P)=u(c^M) v(c^P). The interaction is subspace-identifiable if every other minimal rank-r factorization ~⊤~ u v inducing the same p(xM,xP)p^D(x^M,x^P) satisfies spanu~1,…,u~r=spanu1,…,ur,spanv~1,…,v~r=spanv1,…,vr. [rgb]0,0,0span\ u_1,…, u_r\=span\u_1,…,u_r\, \ v_1,…, v_r\=span\v_1,…,v_r\. Equivalently, the ordered mode basis may change by an invertible transformation in GL(r)GL(r), but the molecule-side and protein-side interaction subspaces are fixed. Assumption 1 (Concept recoverability). The latent concepts Mc^M and Pc^P can be written as functions of the corresponding observations Mx^M and Px^P, respectively. Assumption 2 (Prior independence). Before conditioning on B, the two modalities are independent, i.e., p(M,P)=p(M)p(P)p(c^M,c^P)=p(c^M)p(c^P). Assumption 3 (Positivity / coverage). We have P(B=1)>0P(B=1)>0, and every atomic concept appears with positive probability under the binder-selected distribution p^D. In Definition 1, component-wise identifiability is stronger because it recovers each concept, whereas subspace identifiability recovers only a low-dimensional subspace spanned by several concepts. Assumptions 1–3 are shared by subsequent theorems. They ensure that (i) concept information is preserved from the observed modalities, (i) cross-modal dependence arises only through selection, and (i) sufficient coverage is available for recovery, respectively. More discussion is put in App. A4.2. 2.2 Identifiability Results We next characterize, in a progressively structured way, when the latent interaction mechanism is identifiable from binder-selected data. First, binder-only selection by itself is insufficient: without suitable structural constraints, interacting protein and molecule concepts are not identifiable. Second, such concepts and sparse interactions are component-wise identifiable under proper structural sparsity conditions. Third, low-rank interaction structure yields subspace identifiability when only low-dimensional interaction subspaces can be determined. We give the three formal statements below. Theorem 1 (Non-identifiability without structural constraints). Under the causal model in Section 2.1, suppose Assumptions 1–3 hold. Then even in the bilinear compatibility subclass, for any training distribution p(xM,xP)p^D(x^M,x^P) there exist infinitely many distinct tuples (gM,gP,ℳ,ϕ)(g^M,g^P,M,φ) inducing the same p(xM,xP)p^D(x^M,x^P). Hence, the concepts (M,P)(c^M,c^P) and the interaction mask ℳM are non-identifiable without structural constraints. Theorem 2 (Component-wise identifiability under sparse interaction). Under the causal model above, if the observed binder-selected distribution matches the true joint distribution p(xM,xP)p^D(x^M,x^P), and Assumptions 1–3 hold, and: 2.i (Sparse antichain interaction) There exists a sparsity level s≪KMKPs K_MK_P such that ‖ℳ‖0≤s\|M\|_0≤ s. Moreover, letting Ri R_i :=supp(ℳi,⋅)=j∈[KP]:ℳij≠0, :=supp(M_i,·)=\j∈[K_P]:M_ij≠ 0\, Cj C_j :=supp(ℳ⋅,j)=i∈[KM]:ℳij≠0, :=supp(M_·,j)=\i∈[K_M]:M_ij≠ 0\, the row supports Rii=1KM\R_i\_i=1^K_M and column supports Cjj=1KP\C_j\_j=1^K_P form antichains: Ri⊈Ri′∀i≠i′,Cj⊈Cj′∀j≠j′. [rgb]0,0,0R_i R_i ∀ i≠ i , C_j C_j ∀ j≠ j . 2.i (Sufficient variability) Let η(P,M)≜logp(P∣M)η^D(c^P;c^M) p^D(c^P ^M). There exist molecule-concept values 1,…,R∈int(supp(M))a_1,…,a_R (supp(c^M)), with R≥KMR≥ K_M, such that the vector-valued functions r(P)≜vec([∂2η(P,M)∂ciM∂cjP]i∈[KM],j∈[KP])M=r,r=1,…,R, [rgb]0,0,0 v_r(c^P) \! ( [ ∂^2η^D(c^P;c^M)∂ c_i^M\,∂ c_j^P ]_i∈[K_M],\,j∈[K_P] )_c^M=a_r, r=1,…,R, (4) are linearly independent, where vec(⋅)vec(·) stacks the KMKPK_MK_P mixed-derivative entries. Thus, varying molecule concepts induces enough distinct cross-modal curvature in the selected conditional density. Then, within the stated sparse-antichain model class, the latent concepts (M,P)(c^M,c^P) and the sparse bipartite support of ℳM are component-wise identifiable from p(xM,xP)p^D(x^M,x^P) in the sense of Definition 1. Remark 1. Theorem 1 gives the baseline impossibility result: conditioning on B=1B=1 induces cross-modal concept dependence, but this dependence alone does not determine which concepts or mask generated it. Without structural constraints, infinitely many parameterizations can explain the same distribution; selection is a useful signal but insufficient for identifiability. Theorems 2 and 3 specify what additional structure makes identification possible. Proof is provided in App. A4.4. Remark 2. Theorem 2 shows how the ambiguity in Theorem 1 can be resolved at the component level. Condition 2.i imposes a sparse antichain support: no row support is contained in another row support, and no column support is contained in another column support. This gives each concept a distinct local combinatorial footprint in the interaction graph. Condition 2.i makes these footprints statistically visible: finite molecule-side probes induce linearly independent mixed-derivative response vectors in η(P,M)η^D(c^P;c^M). These conditions identify the individual concepts and sparse interaction support, up to the admissible equivalences in Definition 1. Complete derivation is in App. A4.5. Theorem 3 (Subspace identifiability under low-rank interaction). Under the causal model above, if the observed binder-selected distribution matches the true joint distribution p(xM,xP)p^D(x^M,x^P), and Assumptions 1–3 hold, and: 3.i (Low-rank interaction kernel) Let S(M,P):=∑i=1KM∑j=1KPℳijϕ(ciM,cjP). [rgb]0,0,0S(c^M,c^P):= _i=1^K_M _j=1^K_PM_ijφ(c_i^M,c_j^P). There exists a minimal rank r<minKM,KPr< \K_M,K_P\ and linearly independent mode functions u1,…,ur∈L2(p(M))u_1,…,u_r∈ L^2(p^D(c^M)) and v1,…,vr∈L2(p(P))v_1,…,v_r∈ L^2(p^D(c^P)) such that S(M,P)=∑ℓ=1ruℓ(M)vℓ(P)=(M)⊤(P), [rgb]0,0,0S(c^M,c^P)= _ =1^ru_ (c^M)v_ (c^P)=u(c^M) v(c^P), where =(u1,…,ur)⊤u=(u_1,…,u_r) and =(v1,…,vr)⊤v=(v_1,…,v_r) , and no decomposition with fewer than r separable terms exists. 3.i (Low-rank sufficient variability) There exist probe values 1,…,RM∈int(supp(M))a_1,…,a_R_M (supp(c^M)) and 1,…,RP∈int(supp(P))b_1,…,b_R_P (supp(c^P)), with RM,RP≥rR_M,R_P≥ r, such that the mode-evaluation matrices A:=[u1(1)⋯ur(1)⋱u1(RM)⋯ur(RM)],B:=[v1(1)⋯vr(1)⋱v1(RP)⋯vr(RP)] [rgb]0,0,0U_A:= bmatrixu_1(a_1)&·s&u_r(a_1)\\[-2.84526pt] & & \\[-2.84526pt] u_1(a_R_M)&·s&u_r(a_R_M) bmatrix, _B:= bmatrixv_1(b_1)&·s&v_r(b_1)\\[-2.84526pt] & & \\[-2.84526pt] v_1(b_R_P)&·s&v_r(b_R_P) bmatrix rank(A)=rank(B)=r. [rgb]0,0,0rank(U_A)=rank(V_B)=r. Thus both modality-side mode families can be independently excited across the selected concept support. Then the interaction subspaces M:=spanu1,…,ur⊂L2(p(M)),P:=spanv1,…,vr⊂L2(p(P)) [rgb]0,0,0U_M:=span\u_1,…,u_r\⊂ L^2(p^D(c^M)), _P:=span\v_1,…,v_r\⊂ L^2(p^D(c^P)) are subspace-identifiable from p(xM,xP)p^D(x^M,x^P) in the sense of Definition 1. Remark 3. Theorem 3 states the corresponding subspace-identifiability result. Under Condition 3.i, the interaction kernel is low-rank, so binding is determined by rank-r molecule-side and protein-side interaction subspaces rather than by separately identifiable concept pairs. Condition 3.i is different from Condition 2.i: instead of requiring linearly independent mixed-derivative response vectors, it requires the two mode-evaluation matrices to have full rank. Consequently, the ordered basis of each subspace may change by an invertible transformation, but the rank-r interaction subspaces themselves remain identifiable as invariant statistical objects. Detailed discussion and proof appear in App. A4.6. From theory to implementation. Identifiability matters here because these interaction patterns are reusable and expected to improve generalizability across different scenarios. These results pin down the inductive bias that our method should approximate: (i) concept encoders rich enough to preserve local binding factors, (i) sparse interaction support or low-rank interaction subspaces, and (i) exploitation of the collider-induced dependence under B=1B=1. The next section will translate the proposed theoretical model into concrete neural architectures, with different implementations. The causal formulation enters the implementation through the choice and regularization of an identifiable representation class, rather than through a separate intervention-, propensity-, or counterfactual-based training loss: CausalBind retains the contrastive/ranking supervision of retrieval-based methods, and replaces dense holistic interaction with the sparse-antichain or low-rank parameterizations identified above, together with the corresponding structural regularizer. Accordingly, the empirical gains arise from this theory-derived structural inductive bias, while the causal analysis establishes why these interaction classes, unlike unconstrained dense interactions, are identifiable under binder-only selection; we do not claim that CausalBind estimates interventional effects of individual concepts. 3 CausalBind: Implementation We implement the sparse causal principle of § @sectionsign 2. We develop three variants: CausalBindSP, CausalBindLR, and CausalBindEMB. Due to space, we focus on CausalBindSP, the sparse concept-pair implementation; supplements for CausalBindSP, CausalBindLR, and CausalBindEMB are given in App. A5.2, App. A5.3, and App. A5.4. Inspired by HypSeek (Wang et al., 2026), we use three modalities: molecule structure, pocket structure, and protein sequence, where the sequence branch provides complementary global context beyond local pocket geometry. Instead of aligning holistic embeddings, we first decompose each modality into atomic concepts, and then route protein-molecule matching through sparse concept-level interaction operators (see Figure 3). Notation. Let mol∈ℝNmol×dmolH_mol ^N_mol× d_mol, poc∈ℝNpoc×dpocH_poc ^N_poc× d_poc, and seq∈ℝNseq×dseqH_seq ^N_seq× d_seq denote token representations from Uni-Mol (Zhou et al., 2023) (molecule and pocket) and ESM2 (Lin et al., 2023) (sequence). These project to a per-concept dimension dcd_c. The theory in § @sectionsign 2 uses scalar concepts, corresponding to the special case dc=1d_c=1. In implementation, we allow vector-valued concept tokens with dc≥1d_c≥ 1: for each modality v∈mol,poc,seqv∈\mol,poc,seq\, v∈ℝK×dcC_v ^K× d_c and the i-th row v[i,:]C_v[i,:] represents one concept group. The theory allows different concept counts across modalities (KMK_M and KPK_P); for simplicity, we use a common concept budget K for molecule, pocket, and sequence streams. To align notation with the theoretical mask ℳM, we denote the two implementation masks by ℳmol,pocM_mol,poc and ℳmol,seqM_mol,seq. Their raw learnable parameters are ℳ^mol,poc M_mol,poc and ℳ^mol,seq M_mol,seq, and the effective sparse masks ℳmol,poc,ℳmol,seq∈ℝK×KM_mol,poc,M_mol,seq ^K× K are obtained after activation and thresholding. Pocket3D StructurePocket Encoderpoch_pocConceptExtractorcpoc1c_poc^1cpoc2c_poc^2cpoc3c_poc^3Molecule3D StructureMolecule Encodermolh_molConceptExtractorcmol1c_mol^1cmol2c_mol^2cmol3c_mol^3SequenceProtein SequenceSequence Encoderseqh_seqConceptExtractorcseq1c_seq^1cseq2c_seq^2cseq3c_seq^3111100111100→ Bij=1B_ij=1: Active binding⇢ Bij=0B_ij=0: Blocked Figure 3: Implementation-level view of CausalBind. Each modality-specific encoder maps its input to atomic concepts cvk\c_v^k\, each with dimension dc≥1d_c≥ 1. The indicators BijB_ij at the bottom visualize the branch-wise cross-modal mask ℳa,bM_a,b for (a,b)∈(mol,poc),(mol,seq)(a,b)∈\(mol,poc),(mol,seq)\: Bij=1B_ij=1 when the (i,j)(i,j) entry of ℳa,bM_a,b survives the forward threshold (τ=0.5τ=0.5) (green solid edges, active pair), and Bij=0B_ij=0 otherwise (red dashed edges, blocked pair). Active pairs form the contrastive score, while ℒspaL_spa encourages the interaction sparsity. Concept Extractor. Each modality is decomposed into K latent concepts using a Perceiver-style cross-attention block (Jaegle et al., 2021): ~v(ℓ+1) C^( +1)_v =v(ℓ)+MHA(LN(v(ℓ)),LN(^v),LN(^v)), =C^( )_v+MHA\! (LN(C^( )_v),\,LN( H_v),\,LN( H_v) ), (5) v(ℓ+1) ^( +1)_v =~v(ℓ+1)+FFN(LN(~v(ℓ+1))), = C^( +1)_v+FFN\! (LN( C^( +1)_v) ), where v(0)=v∈ℝK×dcC^(0)_v=Q_v ^K× d_c are learnable concept queries and ^v=vWv∈ℝNv×dc H_v=H_vW_v ^N_v× d_c is the projected token sequence. After L layers, we obtain v∈ℝK×dcC_v ^K× d_c, which serves as the practical counterpart of the latent concepts in § @sectionsign 2. If a backbone exposes only a pooled vector, it corresponds to the special case Nv=1N_v=1; in the main structural branches, token-level features provide nontrivial cross-attention keys and values. Sparsity Mask. Following § @sectionsign 2, we model binding through sparse cross-modal interactions rather than dense alignment. For each branch (a,b)∈(mol,poc),(mol,seq)(a,b)∈\(mol,poc),(mol,seq)\, we compute a,b _a,b =Γ(ℳ^a,b), = ( M_a,b), ℳa,b _a,b =HτST(a,b), =H_τ^ST\! (Z_a,b ), (6) a,b _a,b =ϕ(a,b), =φ(C_a,C_b), ua,b u_a,b =Agg(ℳa,b⊙a,b), =Agg(M_a,b _a,b), where, under the common-K implementation, ℳ^a,b∈ℝK×K M_a,b ^K× K is an unconstrained parameter matrix and ℳa,bM_a,b is the effective sparse mask used for interaction. The activation Γ(ℳ^)=tanh(ℳ^)+1 ( M)= ( M)+1 first produces a continuous mask a,b∈[0,2]K×KZ_a,b∈[0,2]^K× K, and the straight-through threshold operator HτSTH_τ^ST is applied element-wise with mask threshold τ=0.5τ=0.5: the forward pass uses [Hτ(Z)]ij=ZijZij>τ[H_τ(Z)]_ij=Z_ij1\Z_ij>τ\, while the backward pass sends gradients through the continuous ZijZ_ij. Thus thresholding occurs before aggregation, and AggAgg receives only the thresholded masked compatibility matrix ℳa,b⊙a,bM_a,b _a,b. Each nonzero entry ℳa,b[i,j]M_a,b[i,j] represents the strength of interaction between the i-th concept token of modality a and the j-th concept token of modality b. Here ℳa,bM_a,b is the implementation counterpart of the theoretical interaction mask ℳM in § @sectionsign 2. For notational consistency, we use the same compatibility symbol ϕφ: in the scalar theory ϕ:ℝ×ℝ→ℝφ:R×R , while in the vector-valued implementation ϕ:ℝdc×ℝdc→ℝφ:R^d_c×R^d_c . Together, ℳa,bM_a,b and ϕφ instantiate the gated interaction term ℳijϕ(ciM,cjP)M_ijφ(c_i^M,c_j^P) at the token level. The matrix a,b∈ℝK×KA_a,b ^K× K collects all concept-pair compatibilities, where each entry is computed by ϕφ (we use a simple inner product between vector concept tokens). The operator ⊙ denotes element-wise multiplication, and Agg()=K−1∑i,jBijAgg(B)=K^-1 _i,jB_ij aggregates the retained interactions into a branch-level score in our implementation. Thus, the masks ℳmol,pocM_mol,poc and ℳmol,seqM_mol,seq act as cross-modal bottlenecks, enforcing the interaction sparsity. Training Loss. We retain retrieval-based supervision applied after concept decomposition and sparse interaction. The scalar ua,bu_a,b defines the pairwise score for a protein-molecule pair, and over a mini-batch of size N, these scores form matrices a,b∈ℝN×NU^a,b ^N× N. The contrastive loss is ℓcont(a,b)=−12N∑i=1N[logexp(Uiia,b/τc)∑j=1Nexp(Uija,b/τc)+logexp(Uiia,b/τc)∑j=1Nexp(Ujia,b/τc)], [rgb]0,0,0 _cont(U^a,b)=- 12N _i=1^N [ (U^a,b_i/ _c) _j=1^N (U^a,b_ij/ _c)+ (U^a,b_i/ _c) _j=1^N (U^a,b_ji/ _c) ], (7) where τc>0 _c>0 is the contrastive temperature. The combined contrastive objective is ℒcont=ℓcont(poc,mol)+ℓcont(seq,mol). [rgb]0,0,0L_cont= _cont(U^poc,mol)+ _cont(U^seq,mol). (8) The theory models the binder-selected positive distribution; the off-diagonal batch entries here are retrieval negatives used by the contrastive surrogate, not additional observed B=0B=0 labels. We further add sparsity and geometric regularization terms (Wang et al., 2026). The sparsity loss is the entry-wise L1L_1 penalty on the continuous activated masks, which preserves gradients to the raw mask parameters: ℒspa=‖mol,poc‖1+‖mol,seq‖1. [rgb]0,0,0L_spa=\|Z_mol,poc\|_1+\|Z_mol,seq\|_1. (9) The full training objective is ℒ=ℒcont+λspaℒspa+λregℒreg. [rgb]0,0,0L=L_cont+ _spaL_spa+ _regL_reg. (10) Here ℒregL_reg collects the geometric regularizers inherited from HypSeek (Wang et al., 2026). Table 1: Virtual-screening comparison on two benchmarks between baselines and CausalBindSP (K=128,dc=1024K=128,d_c=1024), CausalBindLR (K=64,r=1K=64,r=1), and CausalBindEMB (K=64,dc=256K=64,d_c=256), and CausalBindATOM (K=128,dc=1024K=128,d_c=1024). †S2Drug results are paper-reported (He et al., 2026) rather than reproduced in our pipeline. Method DUD-E (n=102n=102) LIT-PCBA (n=15n=15) AUROC BEDROC80.5 EF1% AUROC BEDROC80.5 EF1% Glide-SP (Friesner et al., 2004) 0.767 0.407 16.18 0.532 0.040 3.41 DeepDTA (Öztürk et al., 2018) 0.584 0.051 2.28 0.563 0.025 1.47 Gnina (McNutt et al., 2021) 0.782 0.299 17.73 0.609 0.054 4.63 RTMScore (Shen et al., 2022) 0.753 0.434 27.10 0.525 0.039 2.94 TankBind (Lu et al., 2022) 0.751 0.330 13.00 0.597 0.039 2.90 BigBind (Brocidiacono et al., 2023) 0.501 0.024 1.18 0.628 0.050 3.79 DrugCLIP (Gao et al., 2023) 0.809 0.505 31.89 0.572 0.062 5.51 GenScore (Shen et al., 2023) 0.816 0.473 28.53 0.596 0.065 5.14 EquiScore (Cao et al., 2024) 0.776 0.432 17.68 0.568 0.049 3.51 DrugHash (Han et al., 2025) 0.837 0.572 37.18 0.546 0.071 6.14 LigUnity (Feng et al., 2025) 0.897 0.674 44.20 0.599 0.075 6.50 HypSeek (Wang et al., 2026) 0.909 0.605 37.83 0.603 0.061 4.62 S2Drug† (He et al., 2026) 0.925 0.793 43.06 0.582 0.087 7.38 CausalBindSP (ours) 0.935 0.744 47.69 0.639 0.097 8.56 CausalBindLR (ours) 0.935 0.704 44.87 0.632 0.103 8.84 CausalBindEMB (ours) 0.939 0.754 48.56 0.636 0.102 9.60 CausalBindATOM (ours) 0.943 0.768 49.66 0.621 0.079 6.25 Instead of letting the contrastive objective operate on dense embeddings, we require it to act through sparse concept-level interactions. This enforces that binding-relevant dependence is captured through a small subset of cross-modal interactions. The other variants change only how the interaction is constrained: CausalBindLR factorizes the raw mask as ℳ^a,b=a,ba,b⊤ M_a,b=U_a,bV_a,b with a,b,a,b∈ℝK×rU_a,b,V_a,b ^K× r before the same activation and gate, CausalBindEMB applies the constrained mask to the pooled embedding instead of the concept axis, and CausalBindATOM feeds the full per-atom Uni-Mol token sequences to the concept extractor. Further implementation details for the main CausalBindSP model, including the explicit form of the geometric regularizers in ℒregL_reg, and a side-by-side comparison of the three main implementations are provided in App. A5.2 and App. A5.5. 4 Experiments Experimental setup. We evaluate on two complementary benchmarks: DUD-E (Mysinger et al., 2012), which covers broad target diversity, and LIT-PCBA (Tran-Nguyen et al., 2020; Mayr et al., 2018), which suppresses property-based shortcuts and is the stricter test of genuine binding discrimination. We report standard metrics: AUROC, BEDROC80.5, and EF@1%, with the latter two emphasizing early enrichment. We adopt the same curated assay-level training dataset as LigUnity (Feng et al., 2025) and HypSeek (Wang et al., 2026), constructed from ChEMBL (Mendez et al., 2019), BindingDB (Gilson et al., 2016), and PDBBind (Liu et al., 2017). As in these baselines, we remove every ChEMBL/BindingDB assay whose target UniProt ID appears in DUD-E, LIT-PCBA, or DEKOIS 2.0, reducing this portion from 26,729 to 18,316 assays, while the PDBBind pairs are not filtered by UniProt ID (App. A5). The evaluation is therefore a cross-dataset generalization protocol rather than a random split of the training pool, with about 476K training molecules and 130K pockets. All models are trained for 50 epochs on four NVIDIA A100 GPUs under the same pipeline for a fair comparison, and the main tables report evaluation results from the last checkpoint, which avoids tuning checkpoint selection to the held-out test benchmarks in the cross-dataset generalization setting. Main Comparison on Virtual Screening. Table 1 compares CausalBind with three families of virtual-screening methods: classical docking systems, structure-aware scoring models on protein-ligand complexes, and retrieval-based representation learners. Retrieval methods are strongest in this large-scale regime, so the key comparison is against recent retrieval baselines. CausalBindSP improves over LigUnity and HypSeek on all six metrics, with the largest gains on LIT-PCBA early enrichment, where property shortcuts are least useful: EF@1% rises from 6.506.50 to 8.568.56 over LigUnity and from 4.624.62 to 8.568.56 over HypSeek, while BEDROC80.5 improves from 0.0750.075 and 0.0610.061 to 0.0970.097. Against the paper-reported results of the concurrent S2Drug (He et al., 2026), which uses a heavier two-stage sequence–structure pipeline, S2Drug attains the best DUD-E BEDROC80.5, while all three CausalBind variants achieve higher DUD-E AUROC and EF@1% and higher values on all three LIT-PCBA metrics; a matched EF@0.5% comparison is given in App. A6.2. CausalBindLR and CausalBindEMB further show that the same constrained-interaction principle remains strong under explicit low-rank and embedding-level implementations. We also report CausalBindATOM, an atom-attentive version of CausalBindSP with the same configuration, whose concept extractor attends over the full per-atom token sequences of the Uni-Mol molecule and pocket encoders rather than their pooled tokens, so that concepts can be traced to individual atoms (App. A6.4). It attains the best DUD-E AUROC and EF@1%, but weaker LIT-PCBA early enrichment than the other variants. Overall, sparse or low-rank interaction helps when holistic alignment confronts nuisance correlations. Table 2: Ablation and diagnostic study on DUD-E and LIT-PCBA. CausalBindSP is the sparse mask, CausalBindLR is the low-rank mask, and CausalBindEMB is the pooled embedding-space mask. −Lspa-L_spa removes the sparsity penalty from CausalBindSP, while +Lspa+L_spa tests an optional sparsity penalty on CausalBindLR without changing its rank-one parameterization. embr=1M_emb^r=1 replaces the pooled embedding mask with a rank-one mask. Rows report mean±std_±\,std over five runs. Method DUD-E (n=102n=102) LIT-PCBA (n=15n=15) AUROC BEDROC80.5 EF1% AUROC BEDROC80.5 EF1% LigUnity (Feng et al., 2025) 0.896 ± 0.005 0.660 ± 0.018 43.14 ± 1.42 0.586 ± 0.010 0.077 ± 0.001 6.46 ± 0.18 HypSeek (Wang et al., 2026) 0.914 ± 0.013 0.690 ± 0.081 44.12 ± 6.14 0.603 ± 0.009 0.067 ± 0.007 5.62 ± 0.71 CausalBindSP 0.939 ± 0.004 0.746 ± 0.004 47.99 ± 0.39 0.635 ± 0.006 0.088 ± 0.007 7.55 ± 0.73 CausalBindSP −Lspa-\,L_spa 0.935 ± 0.004 0.732 ± 0.014 46.93 ± 1.07 0.623 ± 0.005 0.084 ± 0.004 7.17 ± 0.36 CausalBindLR 0.937 ± 0.004 0.732 ± 0.011 46.87 ± 0.91 0.630 ± 0.006 0.085 ± 0.006 7.22 ± 0.31 CausalBindLR +Lspa+L_spa 0.935 ± 0.003 0.727 ± 0.007 46.39 ± 0.58 0.623 ± 0.007 0.083 ± 0.005 7.51 ± 0.94 CausalBindEMB 0.938 ± 0.001 0.752 ± 0.005 48.47 ± 0.44 0.634 ± 0.008 0.101 ± 0.007 9.52 ± 0.96 CausalBindEMB +embr=1+M_emb^r=1 0.723 ± 0.022 0.319 ± 0.013 20.96 ± 0.63 0.539 ± 0.003 0.018 ± 0.001 1.38 ± 0.10 Figure 4: Hyperparameter sweeps for CausalBindSP on DUD-E and LIT-PCBA. Solid lines show BEDROC80.5 and dashed lines show EF@1%; shaded bands mark the selected hyperparameters. Ablation Diagnostics. Table 2 reports both component ablations and implementation diagnostics. For CausalBindSP, removing LspaL_spa weakens early enrichment, suggesting that the concept-pair mask is most useful when regularized toward a simple interaction structure. For CausalBindLR, the rank-one factorization is retained in both rows; adding LspaL_spa has limited effect, so the gain should be attributed to the explicit low-rank parameterization rather than to sparsity regularization. For CausalBindEMB, imposing a rank-one pooled embedding mask sharply degrades performance, showing that rank compression must act directly on the concept axis rather than after pooling. Hyperparameter Study. We sweep K (concept count), dcd_c (concept dimension), λ (sparsity weight λspa _spa), and L (Perceiver depth). Figure 4 and Table 3 show robustness around our setting (K=128,dc=1024,λ=10−4,L=4K=128,d_c=1024,λ=10^-4,L=4), with failures: K=256K=256 over-parameterizes, dc≤256d_c≤ 256 underfits, and λ≥10−2λ≥ 10^-2 collapses both benchmarks. Within the useful regime, λ=10−4λ=10^-4 favors LIT-PCBA and λ=10−3λ=10^-3 favors DUD-E, suggesting that the sparsity level is benchmark-dependent but moderate. Overall, the model benefits from a soft interaction bottleneck, while hard-zero masking does not. The antichain statistics in Table 3 connect these regimes to the theory: the default λ=10−4λ=10^-4 has no hard zeros, so it violates the antichain clause of Condition 2.i and is instead governed by the low-rank regime of Condition 3.i (Theorem 3); λ=10−3λ=10^-3 yields an exact sparse-antichain support with zero violations, matching Condition 2.i (Theorem 2); and λ=10−2λ=10^-2 preserves the antichain property but over-sparsifies the interaction, so the antichain property alone is insufficient for retrieval performance. Table 3: λspa _spa-scan for CausalBindSP with paired cross-modal mask statistics, averaged over the two branches. Small λspa _spa preserves a low-rank regime, while large λspa _spa drives hard-zero collapse. ‘Anti. viol.’ counts antichain violations, i.e., pairs of row (or column) supports in which one contains the other, over 32,51232,512 pairwise tests. λspa _spa DUD-E LIT-PCBA Cross-modal mask statistics AUC BEDROC80.5 EF@1% AUC BEDROC80.5 EF@1% L1L_1 avg %zeros r@95%r@95\% r@99%r@99\% Anti. viol. 00 0.938 0.746 48.14 0.617 0.080 6.83 0.93 0.0 1 25.0 32,512 10−610^-6 0.940 0.749 47.94 0.635 0.088 7.26 0.92 0.0 1 25.0 32,512 10−510^-5 0.938 0.735 47.09 0.607 0.078 6.29 0.88 0.0 1 26.0 32,512 10−410^-4 0.935 0.744 47.69 0.639 0.097 8.56 0.61 0.0 1 34.0 32,512 10−310^-3 0.943 0.768 49.73 0.614 0.073 5.49 0.21 20.5 45.5 79.5 0 10−210^-2 0.437 0.013 0.53 0.457 0.011 0.46 0.06 78.4 74.0 96.0 0 10−110^-1 0.297 0.006 0.30 0.437 0.013 0.59 0.02 92.0 75.5 97.5 1 11 0.725 0.099 5.30 0.595 0.028 1.44 0.01 96.0 75.5 96.5 324 Out-of-distribution generalization and interpretability. We further evaluate zero-shot transfer on DEKOIS 2.0 (Bauer et al., 2013) under target-level and joint target–scaffold out-of-distribution splits, where CausalBind improves over HypSeek and LigUnity on AUROC and BEDROC80.5 in both settings (App. A6.3). An interpretability case study on the Thyroid Hormone Receptor β (THRB) complex from DUD-E further shows that observed binding contacts concentrate within a learned concept pair, and that masking the associated atoms reduces the predicted binding score far more than matched random masks (App. A6.4). Appendix roadmap. For readability, the main text reports only the core comparisons above. Implementation details and reproduction checks are in App. A5, including the three variants CausalBindSP, CausalBindLR, and CausalBindEMB. Additional protocol comparisons, including the best_auc and best_bedroc checkpoints of CausalBindATOM, are in App. A6.1.2, and a longer discussion of the ablation study is in App. A6.1.3. The full hyperparameter tables for K/dcK/d_c, λ, and L are in App. A6.1.4, App. A6.1.5, and App. A6.1.6. The extended theory–practice analysis of the learned masks, including the antichain statistics of Table 3, is in App. A6.1.7. Supplementary DUD-E ROC-enrichment and Merck-FEP affinity-ranking analyses are in App. A6.1.8 and App. A6.1.9, low-rank and embedding-level variant results are in App. A6.5 and App. A6.6, and the full-sweep synthesis is in App. A6.7. The matched EF@0.5% comparison with S2Drug, the DEKOIS 2.0 out-of-distribution evaluation, and the THRB interpretability case study are in App. A6.2, App. A6.3, and App. A6.4. A comprehensive discussion of the remaining theory–practice gap is in App. A6.8. 5 Discussions and Conclusion Limitations, Broader Impact, and Reproducibility. Our evidence is centered primarily on retrieval-based virtual screening under cross-dataset generalization, especially the two main benchmarks, DUD-E and LIT-PCBA, with an additional Merck-FEP-8 affinity-ranking transfer analysis in App. A6.1.9 and target- and scaffold-level out-of-distribution evaluation on DEKOIS 2.0 in App. A6.3. This is the right, deliberately focused scope for the main question of the paper, but it does not yet establish uniform gains across sequence-family or time-stratified splits or larger benchmark collections, and our interpretability analysis provides model-level attribution rather than proof of a physical causal mechanism (see App. A7). As with any virtual-screening method, our model could be misused to prioritize ligands for harmful targets. Although it is a scoring tool rather than a generator, downstream deployment should still be paired with safety and experimental validation filters before any wet-lab follow-up. All source code is publicly available in our GitHub repository to support reproducibility. Conclusion. This paper revisits retrieval-based virtual screening from a structural perspective. Existing methods largely rely on dense shared-space alignment, although protein-molecule binding is typically driven by a small set of local compatibilities. We formalized this mismatch with a V-structure causal model under binder-only selection, showing that unconstrained dense interaction is non-identifiable, while sparse interaction can yield component-wise identifiability and low-rank interaction can yield subspace identifiability. CausalBind turns this principle into a concept-based retrieval framework with three structure-constrained implementations, and consistently improves over strong baselines on DUD-E and LIT-PCBA, with especially clear gains on LIT-PCBA early enrichment. Beyond the two main benchmarks, the learned interaction structures transfer to target- and scaffold-level out-of-distribution splits of DEKOIS 2.0, and the learned masks exhibit the sparse-antichain and low-rank regimes predicted by the theory. Overall, scalable virtual screening should learn not only better shared embeddings but also more reusable cross-modal interaction structures. Acknowledgments We would like to acknowledge the support from NSF Award No. 2229881, AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract R01HL159805, and grants from Quris AI, Florin Court Capital, MBZUAI-WIS Joint Program, and the Al Deira Causal Education project. Meanwhile, we also thank the reviewers for their constructive suggestions. References Bauer et al. (2013) M. R. Bauer, T. M. Ibrahim, S. M. Vogel, and F. M. 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Appendix for “CausalBind: Causal Modeling and Learning for Protein-Molecule Virtual Screening” Table of Contents: Appendix A1 Drug Discovery Pipeline and Method Positioning Figure A1 places CausalBind within the broader drug-discovery workflow. The key point, consistent with the main text, is that our method is designed for the high-throughput virtual-screening stage: the part of the pipeline that must rapidly down-select a very large candidate library before more expensive docking and experimental follow-up. In other words, CausalBind is not a pose-prediction or lead-optimization method; it is a retrieval-style early filter whose job is to improve the quality of the top-ranked compounds passed to later stages. Stage descriptions. The pipeline in Figure A1 is best read as a funnel in which computational cost increases while the candidate set shrinks. It begins with a virtual compound library of roughly 10910^9–101010^10 commercially available or synthetically accessible molecules, such as those represented by ZINC or Enamine REAL. Basic drug-likeness and target-specific filters then reduce this space to a filtered library of roughly 10510^5–10610^6 candidates. The next stage is virtual screening, where methods rank compounds by predicted binding relevance and reduce the candidate set to something tractable for follow-up; this is the stage where CausalBind operates. More expensive docking or rescoring is then applied to a much smaller set, typically 10210^2–10310^3 compounds, using physics-based or structure-aware scoring functions to reason in greater detail about poses and local interactions. Only after this reduction do compounds proceed to experimental validation, usually tens to hundreds of biochemical or cell-based assays, followed by hit-to-lead optimization, and eventually the even narrower stages of clinical testing and approval. The practical role of CausalBind is therefore upstream: it improves which candidates are allowed to reach these far more expensive downstream stages. Virtual Compound Library (ZINC, Enamine REAL) 10910^9–101010^10 Filtered Library (Drug-likeness filters) 10510^5–10610^6 Virtual Screening (Contrastive Learning) CausalBind operates here 10210^2–10410^4This Work Docking / Rescoring (Glide, Vina, ML scoring) 10210^2–10310^3 Experimental Validation (Biochemical assays) 10110^1–10210^2 Hit-to-Lead (Lead optimization) 10010^0–10110^1 Clinical Trials <10<10 Approval ∼1 1 Figure A1: Drug discovery pipeline funnel. Each stage progressively filters compounds, with numbers indicating typical compound counts. Our CausalBind method (green box) operates at the virtual screening stage, improving the quality of top-ranked candidates before expensive downstream validation. By identifying invariant binding determinants and suppressing spurious correlations, our method aims to increase the hit rate in subsequent experimental testing. Why virtual screening matters. The virtual-screening stage is where the paper’s evaluation metrics become practically meaningful. This stage is responsible for the largest candidate reduction before wet-lab testing, so the quality of the very top of the ranking matters far more than average prediction quality over a fixed benchmark. This is exactly why the main text emphasizes early-enrichment metrics such as BEDROC80.5 and EF@1%: even a modest improvement there can substantially reduce the number of false positives advanced to docking or assay validation. Our focus on sparse binding determinants is intended to improve this high-value early-filtering regime. Complementary to docking. This positioning also clarifies why the paper compares docking, structure-aware scoring, and retrieval as distinct regimes rather than as direct substitutes. Docking remains valuable because it reasons explicitly about poses and local geometry, but it is too expensive to serve as the first-pass search over very large libraries. Retrieval-style methods occupy a different operating point: they trade mechanistic detail for scale and are therefore best used upstream of docking. CausalBind fits this role. Its contribution is to improve the representation quality of that pre-filtering stage so that the compounds passed downstream are more likely to contain true binders. Appendix A2 Extended Related Work From docking to scalable screening. Classical virtual screening is dominated by docking pipelines such as Glide, AutoDock Vina, and Surflex, which explicitly search binding poses and rank compounds using physics-inspired scoring functions [Friesner et al., 2004, Trott and Olson, 2010, Spitzer and Jain, 2012]. Their strengths are interpretability and direct geometric reasoning about binding modes, but their cost grows quickly with library size and target flexibility. In practical workflows this often makes docking a downstream stage, applied after an initial coarse filter rather than as the first-pass search over millions of compounds. This is exactly the regime targeted in the main text: large-scale pre-filtering, where retrieval speed and early enrichment matter more than full pose resolution. Learned scoring functions and pose-aware models. A large literature replaces hand-engineered docking scores with learned interaction models over protein-molecule complexes. Gnina introduced deep learning into docking-time scoring with 3D CNNs [McNutt et al., 2021]; RTMScore, GenScore, EquiScore, PIGNet, and PLANET further strengthened this line with graph, transformer, equivariant, and physics-informed architectures [Shen et al., 2022, Shen et al., 2023, Cao et al., 2024, Moon et al., 2022, Zhang et al., 2023]. Related structure-based models such as TankBind and BigBind learn from geometric complexes or large-scale structural supervision to improve affinity prediction and screening robustness [Lu et al., 2022, Brocidiacono et al., 2023]. These methods are highly relevant baselines because they capture detailed intermolecular geometry, but they usually assume a candidate complex, pose, or structure-conditioned interaction graph at inference time. By contrast, our focus is on retrieval-style virtual screening where compound scoring must remain fast enough for large libraries. Affinity prediction beyond screening. Drug–target affinity (DTA) models such as DeepDTA and GraphDTA predict binding strength from sequence or graph representations without necessarily producing an efficient retrieval system [Öztürk et al., 2018, Nguyen et al., 2021]. This line clarified how learned molecular and protein embeddings can support affinity estimation, but it is typically evaluated on regression-style benchmarks rather than target-wise early enrichment. The distinction matters for our paper because the objective is not simply to estimate affinity well on average; it is to identify which compounds should be placed at the very top of the screening list. Retrieval-based contrastive learning for virtual screening. The recent shift most relevant to our work is the move from explicit complex scoring to shared embedding spaces. DrugCLIP [Gao et al., 2023] framed protein-molecule virtual screening as dense retrieval, using Uni-Mol-based encoders and CLIP-style contrastive learning [Radford et al., 2021, Zhou et al., 2023]. This design substantially improved scalability and helped establish retrieval-based virtual screening as a viable alternative to docking, a direction later amplified by a genome-scale Science study built on DrugCLIP-style contrastive screening [Jia et al., 2026]. Subsequent methods enriched this paradigm in different ways: DrugHash compresses embeddings into binary codes for memory-efficient screening [Han et al., 2025]; LigUnity unifies virtual screening with hit-to-lead affinity ranking through scaffold discrimination and pharmacophore ranking in a shared pocket–ligand space [Feng et al., 2025]; HypSeek argues that hyperbolic geometry better captures hierarchical and fine-grained affinity variation than Euclidean retrieval alone [Wang et al., 2026]; and the concurrent S2Drug bridges protein sequence and 3D structure through ChEMBL sequence pretraining and PDBBind fine-tuning with residue-level sequence–structure fusion and an auxiliary binding-site task [He et al., 2026]. These papers collectively show that the field has moved from asking whether retrieval can work at all to asking what kind of structural bias should govern the shared space. The present paper answers that question by shifting the focus from dense holistic alignment to sparse or low-rank cross-modal interaction. Benchmark bias and generalization. A recurring concern in virtual screening is that benchmark artifacts can inflate apparent performance. Hidden biases in DUD-E and related datasets have long been known to create shortcuts based on simple physicochemical cues rather than genuine binding recognition [Chen et al., 2019]. More recent analyses argue that debiasing is essential for robust binding-affinity generalization across targets and chemical series [Graber et al., 2025]. This motivates the paper’s emphasis on LIT-PCBA and on early-enrichment gains under more realistic decoys: the goal is not merely to improve in-distribution ranking, but to learn representations that remain useful when easy shortcuts disappear. Why three regimes? Core computational pattern. Although all three regimes ultimately rank protein-molecule pairs, their inference-time computational pattern differs sharply, and this is what determines their practical scaling behaviour and where their inductive bias has to live. The defining feature of retrieval-based methods is that protein and molecule embeddings are computed independently (a dual-tower architecture); the binding score then reduces to a similarity, typically a dot product, between two pre-computable vectors, exactly as in dense passage retrieval in NLP [Oord et al., 2018, Radford et al., 2021] or image–text retrieval. This pattern is what unlocks very large-scale screening, but it also constrains where cross-modal interaction can be expressed: the interaction has to be encoded into the representation or the similarity function, rather than being computed by a joint pair-input network at inference time. This is precisely where CausalBind intervenes. Instead of leaving the retrieval score as a dense holistic similarity, we impose sparse concept-level interaction inside the retrieval regime itself. Table A1 makes the contrast explicit; Table A2 then drills down to representative methods. Table A1: Core differences between the three virtual-screening regimes. The retrieval regime decouples encoders into two towers, enabling pre-computed library embeddings and millisecond-scale similarity scoring; the other two regimes condition on the (protein, ligand) pair at inference. CausalBindSP stays in the retrieval regime but replaces the dense holistic similarity with a sparse bipartite concept-pair interaction. Physics-based docking Structure-aware ML scoring Retrieval-based learning Inference input (protein, ligand) pair + pose search (protein, ligand) pair, often pose-conditioned protein and ligand encoded independently Binding score physics + empirical scoring on the pair one forward pass through a joint network on the pair dot product (or cosine) of two precomputed vectors Pre-computable embeddings? No (pose-coupled) No (pair-coupled) Yes (dual-tower) Cost per (P, L) at inference seconds–minutes (pose search dominates) ∼ 10–100 ms (one full forward) sub-millisecond (one dot product) Practical library scale 10410^4–10510^5 10510^5–10610^6 –10^8--10^9 Where the inductive bias lives force field + sampling protocol complex graph / 3D-voxel architecture shape of the shared embedding space and the similarity used on it Typical output docked pose + affinity score affinity score (sometimes a pose) ranked retrieval list Where CausalBindSP sits N/A N/A retrieval regime, but with a sparse bipartite similarity over concept pairs instead of a dense one Cross-regime comparison. Building on the regime-level distinctions of Table A1, Table A2 drills down to representative methods within each regime and highlights the design choice that distinguishes CausalBind from prior retrieval baselines: not a new backbone family, but a different structural prior on cross-modal interaction. Table A2: Representative methods in the three virtual-screening regimes discussed in the main text. The table highlights the main design choice of each method and the role it plays in the comparison, with CausalBindSP differing from prior retrieval baselines by imposing a sparse bipartite interaction prior rather than a dense holistic similarity. Method Core Design Relevance Here Physics-based docking systems Glide-SP [Friesner et al., 2004] Hierarchical sampling with empirical scoring function Industry standard; accurate on small candidate sets but expensive at library scale AutoDock Vina [Trott and Olson, 2010] Empirical scoring + iterated local search Open-source, widely used; not designed for ultra-large libraries Structure-aware machine-learning scoring models DeepDTA [Öztürk et al., 2018] Sequence/graph-based affinity prediction without pairwise retrieval design Useful DTA baseline; not optimized for target-wise early enrichment Gnina [McNutt et al., 2021] 3D CNN scoring on docking-grid voxels Strong pose-rescoring; depends on docked poses RTMScore [Shen et al., 2022] Residue-level GNN re-scoring of poses Good DUD-E AUROC; needs upstream pose enumeration GenScore [Shen et al., 2023] Generalist pose generation + rescoring net Unifies pose and score with one network EquiScore [Cao et al., 2024] SE(3)-equivariant GNN on the binding complex Geometric inductive bias on the complex graph PIGNet [Moon et al., 2022] Physics-informed GNN combining empirical + learned terms Hybrid physics/ML score; strong DUD-E early enrichment TankBind [Lu et al., 2022] Trigonometry-aware GNN, end-to-end pose+affinity Joint binding-site prediction and scoring BigBind [Brocidiacono et al., 2023] GNN trained on large-scale curated binding corpus Stresses the role of training-data scale and curation Retrieval-based representation learning DrugCLIP [Gao et al., 2023] CLIP-style contrastive on Uni-Mol encoders Scalable; adopted at genome scale [Jia et al., 2026] DrugHash [Han et al., 2025] Binary-hash retrieval extension of DrugCLIP Memory-efficient at billion-compound scale; weaker PCBA AUROC LigUnity [Feng et al., 2025] Unified screening + hit-to-lead ranking loss Strong BEDROC/EF on both DUD-E and LIT-PCBA HypSeek [Wang et al., 2026] Hyperbolic geometry + cone hierarchy + ESM-2 sequence branch Prior SOTA on DUD-E and LIT-PCBA; geometry-driven inductive bias CausalBindSP (ours) Sparse bipartite concept-pair interaction over Perceiver concepts Theoretically grounded under Heckman selection; best LIT-PCBA early enrichment Detailed positioning. The methods in Table 1 can be read as a progression in where the binding inductive bias is placed. Glide-SP is the classic physics-based reference point: it reasons explicitly about poses and local contacts, but at a computational cost that makes it unsuitable for first-pass screening over very large libraries. DeepDTA occupies a different corner of the design space: it is important historically because it showed that learned molecular and protein representations can support affinity prediction, but it is not a retrieval-oriented screening model and therefore is not optimized for the target-wise early-enrichment setting emphasized in this paper. Among structure-aware learning-based methods, Gnina, RTMScore, GenScore, EquiScore, TankBind, and BigBind all retain a joint pair-input view of the protein-molecule problem. Gnina and RTMScore are closely tied to docked-pose scoring, inheriting much of the pose dependence of classical docking. GenScore and EquiScore strengthen this line with more expressive learned complex representations, while TankBind pushes further toward end-to-end geometric reasoning over binding sites and complexes. BigBind is especially notable for highlighting the importance of large-scale curated structural supervision. These methods are relevant because they capture explicit intermolecular geometry, but their inference-time cost and pair-conditioned architecture make them fundamentally different from the large-scale retrieval regime studied in our main experiments. The retrieval baselines are the closest comparators. DrugCLIP established the modern dense-retrieval formulation for virtual screening and demonstrated that CLIP-style contrastive learning with Uni-Mol encoders can scale to very large libraries. DrugHash preserves that same dense-alignment philosophy while prioritizing storage and retrieval efficiency through binary hashing, at the cost of some predictive fidelity. LigUnity extends dense shared-space retrieval by coupling virtual screening with hit-to-lead ranking signals, which helps its early-enrichment behavior but does not change the fact that the interaction remains holistic in the shared space. HypSeek is the strongest baseline in this family: it enriches retrieval with hyperbolic geometry, cone-hierarchy regularization, and a protein-sequence branch, yielding a powerful geometric inductive bias but still no explicit structural prior on which cross-modal interactions should matter. This is the point at which CausalBind differs. We stay in the retrieval regime and keep the scalability advantages of independently encoded molecule and protein representations, but we change the structural bottleneck through which binding information must pass. Compared with docking or pose-aware scoring, CausalBind does not attempt to replace explicit structural modeling; instead it improves the front-end screening stage that decides which compounds deserve expensive downstream analysis. Compared with DrugCLIP, DrugHash, LigUnity, and HypSeek, we leave the high-level retrieval pipeline intact but modify the representation bottleneck: a shared ConceptExtractor decomposes each modality into atomic concepts, and a sparse cross-modal mask highlights which concept pairs are causally relevant for binding. Semantically, prior retrieval methods ask what inductive bias should structure the embedding space and answer with metric geometry, compression, or training-time objectives. CausalBind instead asks what inductive bias should structure the interaction between molecule and protein representations, and answers with a sparse or low-rank bipartite kernel grounded in the V-structure causal model of Section 2. In that sense, CausalBind is best understood not as a new backbone family, but as an orthogonal structural prior inside retrieval-based virtual screening itself. Causal representation learning. The conceptual basis for our method comes from causal representation learning [Schölkopf et al., 2021, Hyvarinen et al., 2019], especially work showing that multimodal observations and sparse graphical structure can improve identifiability of latent variables [Daunhawer et al., 2023, Kivva et al., 2021]. ConceptAligner [Xie et al., 2025] operationalized this intuition in a multimodal generative setting, while Sun et al. [2025] extended related ideas to biomedical multimodal data with identifiability guarantees for the latent causal variables. We adapt this literature to protein-molecule modeling under a different causal graph: unlike text-to-image generation, neither modality generates the other; instead molecule and protein concepts jointly determine a binding event through a selected collider. This shift from a directional chain to a V-structure is what motivates the sparse bipartite mask at the center of CausalBind. Causal discovery and selection bias. Complementary to causal representation learning, causal discovery recovers causal graphs among observed variables, with recent advances that handle deterministic relations [Li et al., 2024a], decentralized heterogeneous data [Li et al., 2024b], and scalable DAG structure learning [Ziu et al., 2024]. A related line of work studies how selection distorts the observed distribution: causal graphs with latent confounders and post-treatment selection can be characterized and learned from interventional data [Luo et al., 2026b], gene regulatory networks can be inferred in the presence of selection bias [Luo et al., 2026a], and selective mechanisms have also been exploited in text generation [Tang et al., 2025]. Our setting shares this concern with selection: training data contain only observed binding pairs, and CausalBind explicitly models this binder-only selection as conditioning on a collider when deriving which interaction structures remain identifiable. Appendix A3 Biological Observations of Sparse Binding Hot spots and sparse binding energy. A direct biological motivation for our sparse-interaction view comes from the classical hot-spot literature on molecular recognition. In the human growth hormone–receptor system, Clackson and Wells [1995] combined the crystal structure of the complex with alanine-scanning mutagenesis and found that many side chains participate in the physical interface, but the energetic contribution is highly concentrated: a central hydrophobic region on the receptor side, dominated by two tryptophan residues, accounts for more than three quarters of the binding free energy. This result separates the structural epitope, namely the broad set of contacts visible in the complex, from the functional epitope, namely the much smaller subset of contacts that actually maintains affinity. Bogan and Thorn [1998] then showed that this concentration of binding energy is a general property of protein interfaces rather than an isolated example: hot spots are formed by a small subset of residues, are enriched in tryptophan, tyrosine, and arginine, and are often surrounded by less energetic residues that shield the hot spot from bulk solvent. These findings support the central modeling assumption of CausalBind: the binding signal should not be represented as a uniformly dense agreement between whole protein and molecule embeddings. Instead, it is more natural to model binding through a limited number of cross-modal compatibilities, while treating much of the remaining surface or molecular context as supportive, redundant, or potentially nuisance-correlated. Pharmacophores as invariant binding determinants. Even though induced fit allows conformational adjustment, medicinal chemistry has long emphasized that certain feature types remain consistently binding-relevant across conformational states. These are the pharmacophoric features [Wermuth, 2006]: hydrogen-bond donors and acceptors, hydrophobic centers, aromatic groups, charged moieties, and related steric/electronic motifs that repeatedly determine whether a ligand can engage a target pocket productively. The exact geometry of a bound complex may shift, but these feature types remain the stable determinants that make those local interactions possible. In the language of this paper, they are the biologically plausible candidates for the “atomic concepts” that should survive across proteins, ligands, and conformational variants. Why static structures are still informative. A natural concern is that if induced fit is dynamic, then learning from static structures might miss the relevant mechanism. The key point is that our goal is not to recover every conformational degree of freedom; it is to recover the interaction-relevant determinants that persist across such variability. A hydroxyl group remains a hydrogen-bond donor; an aromatic ring remains capable of π-stacking; a hydrophobic patch remains nonpolar even if side chains rotate slightly around it. Benchmarks such as DUD-E and LIT-PCBA are built from static structures, yet successful virtual-screening methods already rely on the fact that these chemically meaningful determinants are more stable than the incidental conformational details. Our method does not remove this assumption; it makes it more explicit by organizing the representation around sparse cross-modal interactions rather than around dense global similarity. From biology to the modeling choice in CausalBind. This biological picture motivates the two central methodological choices in the paper. First, if binding depends on a small set of recurring local determinants, then each modality should be decomposed into smaller concept-like units rather than treated as a single holistic vector. Second, if only a subset of those units actually participates in binding for a given target family, then the cross-modal interaction should itself be sparse or low-rank rather than dense. This is exactly what the ConceptExtractor and SparseMask are designed to implement. The model is therefore not claiming that every learned concept is a textbook pharmacophore; rather, it is imposing an inductive bias that is much closer to the biological mechanism than dense shared-space alignment. Relevance to generalization and drug discovery. This interpretation also clarifies why the largest gains tend to appear on the more bias-controlled LIT-PCBA benchmark. If a method relies on shortcut correlations, it may rank compounds well on easier datasets without truly isolating binding determinants. By contrast, a representation organized around invariant local compatibilities should transfer better to novel targets and novel scaffolds, because those compatibilities are closer to the actual mechanism medicinal chemists reason about during lead optimization. In this sense, the biological motivation of CausalBind is not only explanatory but also practical: it supports the paper’s broader claim that sparse concept-level interaction is a more plausible route to robust virtual screening than dense holistic alignment. Appendix A4 Theoretical Framework: Selection and Identifiability This appendix provides the complete theoretical framework for CausalBind. We cast protein-molecule binding prediction as a Heckman-style selection problem on a V-structure causal graph: the training set consists only of binding pairs (B=1B=1), and our goal is to identify the latent concepts M,Pc^M,c^P and the bipartite interaction mask ℳM from this selection-biased observation. Following the didactic organization of SEDGE [Zhang et al., 2026] and ConceptAligner [Xie et al., 2025], we proceed from a careful SCM formulation (A4.1) through a worked KM=KP=2K_M=K_P=2 toy case (A4.3) to the general non-identifiability and identifiability results (A4.4–A4.5), a low-rank subspace-identifiability result (A4.6), and an explicit comparison to SEDGE (A4.7). Each theorem subsection is organized into a proof sketch, an intuitive example, and a full proof, and a later discussion in the experimental appendix (A6.8) returns to how these identifiability results should be interpreted in light of the learned masks. Our framework borrows the Heckman-selection machinery recently formalized for vision–language concept generation by Xie et al. [2025] and for compositional extrapolation by SEDGE, but differs in four crucial ways: (i) we have a collider causal structure (V-structure) rather than a chain; (i) we have two symmetric observation sources (molecule and protein) rather than one feature vector; (i) our binding kernel is bilinear-sparse rather than a parametric MLP head; and (iv) we replace the SEDGE-style “conservative” assumption (P(S=0)≈0P(S=0)\!≈\!0) with a low-rank binding kernel assumption that is better suited to drug discovery, where most protein-molecule pairs do not bind. A4.1 Structural Causal Model Variables and generative process. Let m∈ℳm and p∈p denote an observed molecule (3D structure) and protein (3D structure). We posit that each modality is generated from latent concepts: M=[c1M,…,cKMM]∈ℝKM,P=[c1P,…,cKPP]∈ℝKP, [rgb]0,0,0c^M=[c^M_1,…,c^M_K_M] ^K_M, ^P=[c^P_1,…,c^P_K_P] ^K_P, (11) through observation maps with modality-specific nuisance noise m=gM(M,εM),p=gP(P,εP), [rgb]0,0,0m=g^M(c^M, ^M), p=g^P(c^P, ^P), (12) where εM ^M and εP ^P capture observation noise, conformational variability, or other modality-specific nuisance factors that are independent of the cross-modal binding event given the concepts. We do not require gMg^M and gPg^P to be globally invertible on the full observation spaces. The recoverability assumption below only requires that the interaction-relevant concepts can be expressed as functions of their corresponding observations on the data support. Biologically, each ciMc^M_i (resp. cjPc^P_j) captures an atomic chemical/structural feature of the molecule (protein), analogous to a pharmacophoric element (hydrogen-bond donor, hydrophobic contact, aromatic ring, etc.). Binding as a collider. The binding indicator B∈0,1B∈\0,1\ is a collider: it is determined jointly by the two concept vectors through a sparse bipartite interaction, B∼Bernoulli(σ(∑i=1KM∑j=1KPℳijϕ(ciM,cjP))), [rgb]0,0,0B \! (σ\! ( _i=1^K_M _j=1^K_PM_ij\,φ(c^M_i,c^P_j) ) ), (13) where ℳ∈ℝ≥0KM×KPM _≥ 0^K_M× K_P is the sparse bipartite binding mask; binary masks and [0,1][0,1] relaxations are special normalizations, while scale is shared with ϕφ. Here KMK_M and KPK_P count concepts, and each atomic concept is scalar in this theoretical model, i.e., ciM,cjP∈ℝc_i^M,c_j^P . Thus ϕ:ℝ×ℝ→ℝφ:R×R maps one molecule concept and one protein concept to a scalar logit-level compatibility score; σ is the logistic. The graph is εM→m←M→B←P→p←εP, [rgb]0,0,0 ^M→ m ^M→ B ^P→ p← ^P, (14) exhibiting a V-structure with B as the collider (Figure 3 in the main text). Bayesian network and conditional-independence structure. We treat (M,P,εM,εP,B,m,p)(c^M,c^P, ^M, ^P,B,m,p) as a Bayesian network [Pearl, 1985, Koller and Friedman, 2009] with respect to the DAG in Eq. (14). The local Markov property yields the following conditional independence (CI) relations, which we use repeatedly in proofs: M⟂P ^M \!\!\! ^P (marginal independence from V-structure), (marginal independence from V-structure), (15) m⟂P,εP,B,p|(M,εM) [rgb]0,0,0m \!\!\! \c^P, ^P,B,p\ (c^M, ^M) (molecule view depends only on molecule concepts and noise), [rgb]0,0,0(molecule view depends only on molecule concepts and noise), (16) p⟂M,εM,B,m|(P,εP) [rgb]0,0,0p \!\!\! \c^M, ^M,B,m\ (c^P, ^P) (protein view depends only on protein concepts and noise), [rgb]0,0,0(protein view depends only on protein concepts and noise), (17) B⟂m,p,εM,εP|(M,P) [rgb]0,0,0B \!\!\! \m,p, ^M, ^P\ (c^M,c^P) (binding mediated entirely by concepts). [rgb]0,0,0(binding mediated entirely by concepts). (18) Heckman selection. The training corpus D contains only positive binding pairs, i.e., samples drawn from p(m,p,M,P)=p(m,p,M,P∣B=1).p^D(m,p,c^M,c^P)\;=\;p\! (m,p,c^M,c^P B=1 ). (19) Equivalently, the full selected joint before marginalizing nuisance variables is p(m,p,M,P,εM,εP)=p(m,p,M,P,εM,εP∣B=1). [rgb]0,0,0p^D(m,p,c^M,c^P, ^M, ^P)=p\! (m,p,c^M,c^P, ^M, ^P B=1 ). (20) In Heckman-selection terminology [Heckman, 1979], B plays the role of the selection variable S: we observe (xM,xP)(x^M,x^P) if and only if B=1B=1. Because B is a collider in Eq. (14), conditioning on B=1B=1 induces the classical explaining-away dependence between Mc^M and Pc^P even though they are marginally independent. Explicitly, using Eq. (15) and Bayes’ rule: p(M,P):=p(M,P∣B=1)=p(M)p(P)P(B=1∣M,P)P(B=1).p^D(c^M,c^P)\;:=\;p(c^M,c^P B=1)\;=\; p(c^M)\,p(c^P)\,P(B=1 ^M,c^P)P(B=1). (21) This is the key handle: the observed joint distribution carries information about (ℳ,ϕ)(M,φ) through the explaining-away factor P(B=1∣M,P)P(B=1 ^M,c^P). Notation. Following SEDGE’s conventions [Zhang et al., 2026], uppercase letters denote random variables and lowercase their realizations. We write p(⋅):=p(⋅∣B=1)p^D(·):=p(· B=1) for distributions implied by the (selection-biased) training data, and reserve p(⋅)p(·) for population distributions (which may involve the unobserved B=0B=0 regime). We use ∝ for equality up to a positive multiplicative constant and 1 for an all-ones vector. Supports and densities are assumed to exist with respect to common dominating measures (typically Lebesgue in ℝKMR^K_M and ℝKPR^K_P, or counting in the corresponding binary spaces). A4.2 Discussion of Assumptions The formal assumptions and theorem-specific conditions are stated in the main text, so we do not repeat them here. Instead, this subsection explains what each assumption or condition contributes to the proofs and how strong it should be read. We keep the main-text notation throughout: Assumptions 1–3 are shared by all theorems, Conditions 2.i–2.i belong to Theorem 2, and Conditions 3.i–3.i belong to Theorem 3. • Assumption 1 (concept recoverability). This assumption says that the concepts entering the binding mechanism are statistically accessible from the observed molecule or protein. It should not be read as requiring a global inverse for the full observation maps gM,gPg^M,g^P or for the neural encoders used in implementation. The proofs use it only to pull the binder-selected observation distribution p(xM,xP)p^D(x^M,x^P) back to concept coordinates on the data support. • Assumption 2 (prior independence). This is the V-structure assumption: before conditioning on binding, molecule-side and protein-side concepts are independent objects. The observed dependence in p^D then comes from conditioning on the collider B=1B=1. Without this assumption, cross-modal dependence could already be present before selection, and the proof could not attribute it to the binding mechanism. • Assumption 3 (positivity / coverage). This is the minimal support requirement. If a concept never appears in binder-selected data, it leaves no statistical trace and cannot be identified by any method. The assumption does not require uniform coverage of all concept combinations; it only rules out concepts that are completely absent from the selected support. • Condition 2.i (sparse antichain interaction). Sparsity alone is not enough for component-wise identifiability: two concepts can still be confounded if one concept’s interaction support is absorbed by another’s. The antichain requirement gives each molecule-side and protein-side concept a distinct combinatorial footprint in the bipartite interaction graph. This is the structural condition that turns the collider-induced dependence into component-wise recovery. • Condition 2.i (sufficient variability). This is the finite-probe richness condition for the sparse theorem. The mixed derivatives of η(P,M)=logp(P∣M)η^D(c^P;c^M)= p^D(c^P ^M) are the statistical objects that survive after marginal terms are removed in the proof. Requiring the response vectors at finitely many molecule-side probes to be linearly independent rules out degenerate selected distributions in which different interaction patterns induce the same curvature. • Condition 3.i (low-rank interaction kernel). This is the structural condition for the subspace theorem. It replaces exact concept-pair support recovery with the assumption that the binding score has a minimal rank-r separable factorization. Thus the target of identification changes: we no longer seek a unique entry-wise mask, but the molecule-side and protein-side subspaces spanned by the low-rank interaction functions. • Condition 3.i (low-rank sufficient variability). This is the statistical richness condition for the subspace theorem, not another low-rank constraint. It requires the mode-evaluation matrices to have full rank at finitely many molecule-side and protein-side probes. This guarantees that the low-rank modes specified by Condition 3.i are sufficiently excited on the selected support, so their spans can be recovered from the observed distribution. A4.3 Toy Case: Two Concepts Before the general theorem, we work out the simplest non-trivial case explicitly, following SEDGE’s didactic strategy. Assume KM=KP=2K_M=K_P=2 with binary concepts c1M,c2M,c1P,c2P∈0,1c^M_1,c^M_2,c^P_1,c^P_2∈\0,1\, so each concept is either “active” or “inactive.” The binding kernel is f(M,P):=∑i=12∑j=12ℳijciMcjP.f(c^M,c^P)\;:=\; _i=1^2 _j=1^2M_ij\,c^M_i\,c^P_j. (22) and P(B=1∣M,P)=σ(f(M,P))P(B=1 ^M,c^P)=σ(f(c^M,c^P)). Case (a): Diagonal mask (identifiable). Suppose ℳdiagM^diag is the 2×22× 2 identity matrix. Binding requires ciM=ciP=1c^M_i=c^P_i=1 for some i. This mask satisfies Condition 2.i: rows are 1\1\ and 2\2\, forming an antichain. Using Eqs. (21) and (15), the selected joint factorizes across the two “channels”: p(M,P) p^D(c^M,c^P) ∝p(c1M)p(c2M)p(c1P)p(c2P)σ(c1Mc1P+c2Mc2P) p(c^M_1)p(c^M_2)\,p(c^P_1)p(c^P_2)\,σ(c^M_1c^P_1+c^M_2c^P_2) ≈p(c1M,c1P)⏟channel 1⋅p(c2M,c2P)⏟channel 2, ≈ p^D(c^M_1,c^P_1)_channel 1· p^D(c^M_2,c^P_2)_channel 2, (23) where the factorization is only a didactic approximation, not an identity of the logistic selection model. The formal theorem below does not rely on this approximation; it uses mixed-derivative responses of the selected density instead. Each channel is identifiable from training data independently: p(ciM,ciP)p^D(c^M_i,c^P_i) is directly observed, and the marginals p(ciM),p(ciP)p(c^M_i),p(c^P_i) are recovered via standard deconvolution using the selection equation. Concept labels (1,2)(1,2) remain permutable, an indeterminacy that is included in the equivariance class of Theorem 2. Case (b): Dense mask (non-identifiable). Suppose ℳdenseM^dense is the 2×22× 2 all-ones matrix. Then f(M,P)=(c1M+c2M)(c1P+c2P)f(c^M,c^P)=(c^M_1+c^M_2)(c^P_1+c^P_2), which depends only on the sums sM:=c1M+c2Ms^M:=c^M_1+c^M_2 and sP:=c1P+c2Ps^P:=c^P_1+c^P_2. Consequently, any reparameterization (c~1M,c~2M)( c^M_1, c^M_2) that preserves sMs^M (and similarly for Pc^P) yields the same observed p^D. For instance, the swap (c~1M,c~2M)=(c2M,c1M)( c^M_1, c^M_2)=(c^M_2,c^M_1) with ℳ~=ℳdense M=M^dense is an exact equivalent. More generally, for any T∈GL(2)T (2) satisfying T⊤ℳdenseT=ℳdenseT M^denseT=M^dense (the stabilizer is an infinite continuous group), the transformation preserves the observation. Hence, the latent concepts are not identifiable. Condition 2.i fails because both rows have the same support 1,2\1,2\, which is a trivial antichain of size 11. Takeaway. The toy case isolates the identifying mechanism: a sparse, antichain interaction structure prevents concept-mixing reparameterizations and factorizes the observation into independently identifiable “channels.” Dense (or row-absorbed) masks admit infinite equivalent solutions, providing an intuition for Theorem 1. A4.4 Theorem 1: Non-Identifiability Without Structural Constraints Before establishing positive identifiability, we show that structural constraints on the interaction mask are indispensable: without them, the latent concepts and mask are not identifiable from the training distribution p(xM,xP)p^D(x^M,x^P), even under Assumptions 1–3. Theorem 1 (Non-identifiability without structural constraints). Suppose Assumptions 1–3 hold but no structural constraint is imposed on ℳM. Then, even in the bilinear compatibility subclass, for any training distribution p(xM,xP)p^D(x^M,x^P) there exist infinitely many distinct tuples (gM,gP,ℳ,ϕ)(g^M,g^P,M,φ) all inducing the same p(xM,xP)p^D(x^M,x^P). Hence, the concepts (M,P)(c^M,c^P) and the interaction mask ℳM are non-identifiable without structural constraints. Discussion. This theorem is the appendix counterpart of Theorem 1 in the main text. Without structural restrictions on ℳM, the latent coordinate systems remain unconstrained. The resulting ambiguity is exactly the large GL(KM)×GL(KP)GL(K_M)×GL(K_P) reparameterization orbit discussed in Section 2.2: the same training distribution can be explained by infinitely many equivalent latent coordinate systems. A4.4.1 Proof Sketch Fix any valid tuple (gM,gP,ℳ,ϕ)(g^M,g^P,M,φ) consistent with p^D. In the proof only, let eMe^M and ePe^P denote auxiliary concept-recovery functions witnessing Assumption 1. For any pair of invertible linear maps TM∈GL(KM)T_M (K_M) and TP∈GL(KP)T_P (K_P), set ~M=TMM c^M=T_Mc^M and ~P=TPP c^P=T_Pc^P, and define g~M(,εM) g^M(c, ^M) :=gM(TM−1,εM), :=g^M(T_M^-1c, ^M), e~M(xM) e^M(x^M) :=TMeM(xM), :=T_Me^M(x^M), g~P(,εP) g^P(c, ^P) :=gP(TP−1,εP), :=g^P(T_P^-1c, ^P), e~P(xP) e^P(x^P) :=TPeP(xP), :=T_Pe^P(x^P), ℳ~ M :=TM−TℳTP−1, :=T_M^-TM\,T_P^-1, ϕ~ φ :=ϕ. :=φ. Then the binding score in (13) is preserved: ∑i,jℳ~ijϕ(c~iM,c~jP)=∑i,jℳijϕ(ciM,cjP), _i,j M_ijφ( c^M_i, c^P_j)= _i,jM_ijφ(c^M_i,c^P_j), so the selection factor is unchanged. The transformed recovery functions show that Assumption 1 remains satisfied, and the nuisance variables enter only through the reparameterized observation maps. After integrating those variables out, the reparameterized tuple induces the same training distribution p(xM,xP)p^D(x^M,x^P). Since GL(KM)×GL(KP)GL(K_M)×GL(K_P) is an infinite group, infinitely many distinct transformations yield the same p^D. Critically, these transformed masks generally violate the structural restrictions required for positive recovery, so the ambiguity is ruled out only after such restrictions are imposed. A4.4.2 Intuitive Example The dense case in App. A4.3 is the smallest example. With a 2×22× 2 all-ones mask and ϕ(x,y)=xyφ(x,y)=xy, the binding score becomes S(M,P)=(c1M+c2M)(c1P+c2P). [rgb]0,0,0S(c^M,c^P)=(c^M_1+c^M_2)(c^P_1+c^P_2). The selected distribution therefore sees only the molecule-side sum and the protein-side sum, not the two coordinates separately. Different latent coordinates that preserve these sums, or more generally invertible reparameterizations absorbed by a transformed mask, produce the same binder-selected observations. This is why dense holistic interaction leaves the concepts and mask underdetermined. A4.4.3 Full Proof of Theorem 1 Proof. We construct an explicit infinite family of tuples all producing the same p^D. Fix any reference tuple (gM,gP,ℳ,ϕ)(g^M,g^P,M,φ) compatible with p^D, and let eM,ePe^M,e^P be auxiliary functions witnessing Assumption 1 on the data support. For clarity, consider the bilinear case ϕ(x,y)=xyφ(x,y)=xy; the argument extends to compatibility functions that are locally linear in each argument on the relevant support. Pick any (TM,TP)∈GL(KM)×GL(KP)(T_M,T_P) (K_M)×GL(K_P). Set ~M=TMM c^M=T_Mc^M and ~P=TPP c^P=T_Pc^P, and define the reparameterized tuple: g~M(~M,εM) g^M( c^M, ^M) :=gM(TM−1~M,εM), :=g^M(T_M^-1 c^M, ^M), g~P(~P,εP) g^P( c^P, ^P) :=gP(TP−1~P,εP), :=g^P(T_P^-1 c^P, ^P), ℳ~ M :=TM−⊤ℳTP−1. :=T_M^- \,M\,T_P^-1. (24) The corresponding auxiliary recovery functions are e~M(xM):=TMeM(xM) e^M(x^M):=T_Me^M(x^M) and e~P(xP):=TPeP(xP) e^P(x^P):=T_Pe^P(x^P), so the reparameterized tuple also satisfies Assumption 1. Setting ~M=TMM c^M=T_Mc^M and ~P=TPP c^P=T_Pc^P: ∑i,jℳ~ijc~iMc~jP=~M⊤ℳ~~P=(TMM)⊤TM−⊤ℳTP−1(TPP)=M⊤ℳP. _i,j M_ij c^M_i c^P_j= c^M M c^P=(T_Mc^M) T_M^- MT_P^-1(T_Pc^P)=c^M Mc^P. Hence the binding probability P(B=1∣M,P)P(B=1 ^M,c^P) is identical for the two tuples. Therefore the selected joint over observations, with nuisance variables integrated out, is identical: p(xM,xP)=∬p(xM∣M)p(xP∣P)p(M,P)dMdP, [rgb]0,0,0p^D(x^M,x^P)= p(x^M ^M)p(x^P ^P)p^D(c^M,c^P)\,dc^M\,dc^P, where p(xM∣M)=∫p(xM∣M,εM)p(εM)dεMp(x^M ^M)= p(x^M ^M, ^M)p( ^M)\,d ^M and analogously for xPx^P. The same expression is obtained under the transformed tuple, so p~(xM,xP)=p(xM,xP) p^D(x^M,x^P)=p^D(x^M,x^P). Since GL(KM)×GL(KP)GL(K_M)×GL(K_P) is a connected Lie group of dimension KM2+KP2K_M^2+K_P^2, this parameterizes an uncountable family of equivalent tuples. For generic choices of (TM,TP)(T_M,T_P), the transformed coordinates mix the original concepts and the transformed mask ℳ~=TM−⊤ℳTP−1 M=T_M^- MT_P^-1 does not preserve the structural class required for positive recovery. Thus, without structural restrictions on ℳM, the reparameterizations cannot be distinguished by p^D, and the latent concepts and interaction mask are non-identifiable. Imposing additional structure breaks this argument by ruling out generic coordinate-mixing transformations. ∎ A4.5 Theorem 2: Component-Wise Identifiability Under Sparse Interaction We now state the central identifiability result. Our proof strategy adapts the component-wise identification arguments of Kivva et al. [2021] and Xie et al. [2025] to the V-structure setting, using the collider-induced dependence from Assumption 2 plus conditioning on B=1B=1 as the information source and the sparse antichain structure in Condition 2.i as the disentangling constraint. Theorem 2 (Component-wise identifiability under sparse interaction). Suppose Assumptions 1–3 and Conditions 2.i–2.i of Theorem 2 hold, and let p(xM,xP)=p(xM,xP∣B=1)p^D(x^M,x^P)=p(x^M,x^P B=1) be the observed selection-biased distribution. Then, within the stated sparse-antichain model class, the latent concepts (M,P)(c^M,c^P) and the bipartite support of ℳM are component-wise identifiable from p^D in the sense of Definition 1. Discussion. This is the appendix counterpart of Theorem 2 and the main positive result of the paper. Condition 2.i provides the structural separation that sparse recovery needs: without an antichain support pattern, one concept can still be absorbed into another. Condition 2.i provides the complementary finite-probe richness condition: the mixed-derivative response vectors r(P) v_r(c^P) induced by finitely many molecule-side concept values must be linearly independent. These vectors are exactly the statistical objects used below to separate distinct interaction patterns after the marginal terms have been removed by cross differentiation. The result is a conditional sufficiency statement for the sparse-antichain class: generic dense GLGL coordinate changes are not admissible competitors because they usually destroy this support structure. The theorem therefore isolates the exact point at which collider-induced dependence becomes informative rather than merely confounded. A4.5.1 Proof Sketch We outline the main three steps of the proof. Step 1: Selection-induced factorization. Under Assumption 2, the prior factorizes p(M,P)=p(M)p(P)p(c^M,c^P)=p(c^M)p(c^P). The nuisance terms are modality-specific and are marginalized into the observation kernels p(xM∣M)p(x^M ^M) and p(xP∣P)p(x^P ^P), while Assumption 1 lets us pull the observed selected distribution back to concept coordinates. Combined with Assumption 1 and the selection (19), p(M,P)=p(M,P∣B=1)∝p(M)p(P)P(B=1∣M,P).p^D(c^M,c^P)\;=\;p(c^M,c^P B=1)\; \;p(c^M)\,p(c^P)\,P(B=1 ^M,c^P). (25) The last factor is fully determined by (ℳ,ϕ)(M,φ) via (13). Thus the observed joint over concepts in D is the product of the marginals times a single bipartite coupling term. Step 2: Sparse structure disentangles. Suppose two tuples (gM,gP,eM,eP,ℳ,ϕ)(g^M,g^P,e^M,e^P,M,φ) and (g~M,g~P,e~M,e~P,ℳ~,ϕ)( g^M, g^P, e^M, e^P, M,φ) both induce p^D after marginalizing εM,εP ^M, ^P. From (25), matching the selected joint implies that the coupling term must match up to the marginal factors, which gives ∑i,jℳijϕ(ciM,cjP)=∑i,jℳ~ijϕ(c~iM,c~jP)+Δ(M,P), _i,jM_ij\,φ(c^M_i,c^P_j)= _i,j M_ij\,φ( c^M_i, c^P_j)+ (c^M,c^P), (26) for some residual Δ that depends only on marginal structure. Under Condition 2.i (antichain rows/columns), the rows of ℳM cannot be expressed as linear combinations of subsets of each other. Combined with the finite-probe sufficient variability in Condition 2.i, the mixed-derivative response vectors rr=1R\ v_r\_r=1^R give enough independent equations to rule out the linear reparameterization used in Theorem 1. This forces the recovered concept coordinates induced by (g~M,e~M)( g^M, e^M) and (g~P,e~P)( g^P, e^P) to differ from those induced by (gM,eM)(g^M,e^M) and (gP,eP)(g^P,e^P) only by a component-wise transformation [Kivva et al., 2021]. Step 3: Component-wise identification. Applying the component-wise identification theorem of Kivva et al. [2021], Xie et al. [2025] (whose hypotheses are implied by Assumptions 1–3 and Conditions 2.i–2.i), we obtain that Mc^M is identified up to permutation and element-wise invertible transformation, and symmetrically for Pc^P. The mask ℳM is then identified up to the corresponding permutation of its rows/columns. The full derivation, including the mixed-derivative argument and the reduction from general invertible maps to permutation plus component-wise transformations, is given below. A4.5.2 Intuitive Example The diagonal case in App. A4.3 gives the cleanest component-wise picture. If ℳ=[1001], [rgb]0,0,0M= bmatrix1&0\\ 0&1 bmatrix, then molecule concept c1Mc^M_1 leaves a different interaction footprint from c2Mc^M_2: their row supports are 1\1\ and 2\2\. Conditioning on B=1B=1 therefore creates two distinguishable cross-modal dependence channels, one for each concept pair. By contrast, if the row supports were 1\1\ and 1,2\1,2\, the first footprint would be contained in the second, so the effect of one concept could be partially absorbed into another. This is the intuition behind Condition 2.i: component-wise recovery requires sparse supports that are not nested inside each other. A4.5.3 Full Proof of Theorem 2 Proof. Let (gM,gP,eM,eP,ℳ,ϕ)(g^M,g^P,e^M,e^P,M,φ) be the ground-truth tuple and (g~M,g~P,e~M,e~P,ℳ~,ϕ~)( g^M, g^P, e^M, e^P, M, φ) any other tuple that also produces p^D after marginalizing εM,εP ^M, ^P. By Assumption 1, concepts are identified with recovery-map outputs: M:=eM(xM)c^M:=e^M(x^M) and ~M:=e~M(xM) c^M:= e^M(x^M). For any nuisance value εM ^M in the support, this induces ~M=F(M) c^M=F(c^M) with F(M):=e~M(gM(M,εM))F(c^M):= e^M(g^M(c^M, ^M)); Assumption 1 makes this expression independent of the particular nuisance realization on the data support. Because the reverse map M=eM(g~M(~M,ε~M))c^M=e^M( g^M( c^M, ^M)) maps ~M c^M back to Mc^M on the same support, F is an on-support smooth bijection. Similarly, ~P=G(P) c^P=G(c^P) for an on-support smooth bijection G(P):=e~P(gP(P,εP))G(c^P):= e^P(g^P(c^P, ^P)). Step 1: Derive the matching condition. Using Eq. (21), matching p(M,P)p^D(c^M,c^P) on both sides (with appropriate Jacobians from the F,GF,G reparameterizations) gives logp(M)+logp(P)+logσ(∑ijℳijϕ(ciM,cjP))=logp(F(M))+logp(G(P))+log|detJF|+log|detJG|+logσ(∑ijℳ~ijϕ~(Fi(M),Gj(P)))+const. p(c^M)+ p(c^P)+ σ ( _ijM_ijφ(c^M_i,c^P_j) )=\\ p(F(c^M))+ p(G(c^P))+ | J_F|+ | J_G|\\ + σ ( _ij M_ij φ(F_i(c^M),G_j(c^P)) )+const. (27) Every term on the first line is evaluated in the original coordinates (M,P)(c^M,c^P), while the second line is expressed in the transformed coordinates (F(M),G(P))(F(c^M),G(c^P)). Equality holds pointwise because both tuples induce the same selected density. Step 2: Twice-differentiation trick. Taking ∂2/∂cMi∂cPj∂^2/∂ c^M_i∂ c^P_j of Eq. (27) eliminates all terms that depend only on Mc^M or only on Pc^P. Specifically, ∂2∂ciM∂cjPlogp(M)=0,∂2∂ciM∂cjPlogp(P)=0, ∂^2∂ c_i^M∂ c_j^P p(c^M)=0, ∂^2∂ c_i^M∂ c_j^P p(c^P)=0, and the same is true for the Jacobian terms log|detJF| | J_F| and log|detJG| | J_G|, because each depends on only one modality. We are therefore left with ∂2∂ciM∂cjP[logσ(∑klℳklϕ(ckM,clP))−logσ(∑klℳ~klϕ~(Fk(M),Gl(P)))]=0. ∂^2∂ c^M_i∂ c^P_j [ σ ( _klM_klφ(c^M_k,c^P_l) )- σ ( _kl M_kl φ(F_k(c^M),G_l(c^P)) ) ]=0. (28) Write S(M,P)=∑klℳklϕ(ckM,clP),S~(M,P)=∑klℳ~klϕ~(Fk(M),Gl(P)).S(c^M,c^P)= _klM_klφ(c^M_k,c^P_l), S(c^M,c^P)= _kl M_kl φ(F_k(c^M),G_l(c^P)). Using σ′(x)/σ(x)=1−σ(x)σ (x)/σ(x)=1-σ(x) and the chain rule, we obtain ∂2∂ciM∂cjPlogσ(S) ∂^2∂ c_i^M∂ c_j^P σ(S) =(1−σ(S))∂2S∂ciM∂cjP−σ(S)(1−σ(S))∂S∂ciM∂S∂cjP, =(1-σ(S)) ∂^2S∂ c_i^M∂ c_j^P-σ(S)(1-σ(S)) ∂ S∂ c_i^M ∂ S∂ c_j^P, (29) ∂2∂ciM∂cjPlogσ(S~) ∂^2∂ c_i^M∂ c_j^P σ( S) =(1−σ(S~))∂2S~∂ciM∂cjP−σ(S~)(1−σ(S~))∂S~∂ciM∂S~∂cjP, =(1-σ( S)) ∂^2 S∂ c_i^M∂ c_j^P-σ( S)(1-σ( S)) ∂ S∂ c_i^M ∂ S∂ c_j^P, (30) where the derivatives of S~ S further expand through F and G; for example, ∂S~∂ciM=∑k,lℳ~kl∂1ϕ~(Fk,Gl)∂Fk∂ciM, ∂ S∂ c_i^M= _k,l M_kl\, _1 φ(F_k,G_l)\, ∂ F_k∂ c_i^M, and similarly for ∂S~/∂cjP∂ S/∂ c_j^P and the mixed derivative. Equivalently, these surviving terms form the mixed-derivative matrix of η(P,M)η^D(c^P;c^M). Evaluating the equality at the probe values M=rc^M=a_r from Condition 2.i gives a finite system determined by the response vectors r(P) v_r(c^P). Since 1,…,R v_1,…, v_R are linearly independent, no nonzero linear combination of these cross-modal curvature responses can vanish p(P)p^D(c^P)-almost everywhere; hence the surviving mixed-derivative terms must match component-wise for the interaction patterns under comparison. The important point is that only the cross-modal interaction survives; all purely marginal information has been removed. Step 3: Antichain structure implies permutation + diagonal. The matching condition from Step 2 yields a linear system relating the nonzero pattern of ℳM (at positions (i,j)(i,j)) to linear combinations of entries of ℳ~ M through JFJ_F and JGJ_G. We now invoke the following sparsity-identifiability lemma, adapted from Kivva et al. [2021], Lemma 3, and Lachapelle et al. [2022], Theorem 2: Lemma 1 (Antichain recovery). Let ℳ∈0,1KM×KPM∈\0,1\^K_M× K_P have antichain row and column support as in Condition 2.i. Suppose ℳ~∈ℝKM×KP M ^K_M× K_P, A∈ℝKM×KMA ^K_M× K_M is invertible, and B∈ℝKP×KPB ^K_P× K_P is invertible, and they satisfy ℳ~ij=∑k,lAikℳklBlj,∀(i,j)∈supp(ℳ~). M_ij= _k,lA_ikM_klB_lj, ∀(i,j) ( M). Then A and B are both products of a permutation matrix and a positive diagonal matrix. This lemma says that antichain patterns are preserved only by permutation and positive rescaling. Combined with Step 2, it forces JFJ_F and JGJ_G to be diagonal up to permutation. Equivalently, F=ΠM∘HMF= _M H^M and G=ΠP∘HPG= _P H^P where HM,HPH^M,H^P act component-wise. Intuitively, if one coordinate of F mixed two source concepts, then one support pattern would become expressible through another, contradicting the antichain condition. Step 4: Component-wise monotone identification. With F,GF,G reduced to permutation plus component-wise maps, the remaining question is whether the individual maps hiM,hjPh^M_i,h^P_j are identifiable. Applying the component-wise identification theorem of Kivva et al. [2021] (which relies on Condition 2.i’s finite-probe sufficient variability and Assumption 1), we conclude that each hiMh^M_i is identified up to a strictly monotone invertible transformation of its input, and symmetrically for hjPh^P_j. Step 5: Combining. The freedom remaining is: (i) the permutations ΠM,ΠP _M, _P (concept label swapping), (i) the component-wise monotone maps hiM,hjPh^M_i,h^P_j, and (i) the trivial joint rescaling (ℳ,ϕ)↦(cℳ,c−1ϕ)(M,φ) (cM,c^-1φ) for c>0c>0 that preserves ∑ijℳijϕ(⋅) _ijM_ijφ(·). These are exactly the allowed equivalences in the component-wise identifiability part of Definition 1. ∎ ∎ A4.6 Theorem 3: Subspace Identifiability Under Low-Rank Interaction Condition 2.i requires that some sparse mask exist; it does not specify its structure. For protein-molecule binding, we argue that the subspace-identifiability alternative in Condition 3.i is also natural empirically: the binding kernel has low rank. Motivation. Protein–molecule binding in drug discovery is dominated by a small number of pharmacophoric motifs: hydrogen bonds, π-stacking, hydrophobic packing, and electrostatic complementarity. It is therefore physically reasonable that the complete binding response can be decomposed into r independent interaction modes, each characterized by a pair (uk,vk)(u_k,v_k) describing how each modality contributes to mode k. We expect r to be small; typical estimates from medicinal chemistry place r at 55–2020 for a given target family. Why this replaces SEDGE’s conservative assumption. SEDGE’s identifiability argument relies on P(S=0)≈0P(S=0)≈ 0, i.e., “unseen configurations are rare.” This is false for binding: most protein-molecule pairs do not bind, so P(B=0)≫P(B=1)P(B=0) P(B=1). Condition 3.i replaces this with a physics-motivated structural constraint on the kernel itself, preserving identifiability without requiring the pathological SEDGE-style coverage condition. Architectural implication. Under Condition 3.i, the bilinear binding score can be implemented with a rank-r factorized mask ℳ=UV⊤M=UV where U∈ℝKM×rU ^K_M× r and V∈ℝKP×rV ^K_P× r, reducing parameters from KMKPK_MK_P to r(KM+KP)r(K_M+K_P) and naturally enforcing the low-rank prior. This is the structural basis for our low-rank concept-basis implementation in App. A5.3; the main concept-pair implementation in App. A5.2 can also reach this regime empirically through regularization-induced low rank. Subspace identifiability under low-rank interaction. When Conditions 2.i–2.i are replaced by Conditions 3.i–3.i, the full concept-pair mask need not be identifiable. However, an analog of Theorem 2 still holds at the level of the rank-r interaction subspaces: Theorem 3 (Subspace identifiability under low-rank interaction). Suppose Assumptions 1–3 and Conditions 3.i–3.i of Theorem 3 hold, so the binding score admits a minimal rank-r separable factorization S(M,P)=∑ℓ=1ruℓ(M)vℓ(P)=(M)⊤(P),r<minKM,KP, [rgb]0,0,0S(c^M,c^P)= _ =1^ru_ (c^M)v_ (c^P)=u(c^M) v(c^P), r< \K_M,K_P\, with linearly independent mode families uℓ=1r⊂L2(p(M))\u_ \_ =1^r⊂ L^2(p^D(c^M)) and vℓ=1r⊂L2(p(P))\v_ \_ =1^r⊂ L^2(p^D(c^P)). Assume the low-rank sufficient-variability condition: there exist probe values 1,…,RM∈int(supp(M))a_1,…,a_R_M (supp(c^M)) and 1,…,RP∈int(supp(P))b_1,…,b_R_P (supp(c^P)), with RM,RP≥rR_M,R_P≥ r, such that A:=[u1(1)⋯ur(1)⋱u1(RM)⋯ur(RM)]andB:=[v1(1)⋯vr(1)⋱v1(RP)⋯vr(RP)] [rgb]0,0,0U_A:= bmatrixu_1(a_1)&·s&u_r(a_1)\\[-2.84526pt] & & \\[-2.84526pt] u_1(a_R_M)&·s&u_r(a_R_M) bmatrix _B:= bmatrixv_1(b_1)&·s&v_r(b_1)\\[-2.84526pt] & & \\[-2.84526pt] v_1(b_R_P)&·s&v_r(b_R_P) bmatrix rank(A)=rank(B)=r. [rgb]0,0,0rank(U_A)=rank(V_B)=r. Then the molecule-side and protein-side interaction subspaces M:=spanu1,…,ur⊂L2(p(M)),P:=spanv1,…,vr⊂L2(p(P)) [rgb]0,0,0U_M:=span\u_1,…,u_r\⊂ L^2(p^D(c^M)), _P:=span\v_1,…,v_r\⊂ L^2(p^D(c^P)) are subspace-identifiable from p(xM,xP)p^D(x^M,x^P) in the sense of Definition 1. Discussion. This is the appendix counterpart of Theorem 3. The two theorem-specific conditions play different roles: • Condition 3.i is the structural low-rank condition. It weakens exact entry-wise sparsity into a mode-level prior, so the identifiable object is the pair of low-dimensional interaction subspaces rather than each individual concept pair. • Condition 3.i is the sufficient-variability condition for those low-rank modes. It ensures that the selected data excite the mode-evaluation matrices with full rank, which makes the subspaces statistically recoverable. Together, these conditions define the subspace-identifiability regime relevant to our experiments, where the learned masks are empirically low-rank rather than exactly sparse. A4.6.1 Proof Sketch As in the preceding proofs, nuisance variables εM,εP ^M, ^P are marginalized into the observation kernels and Assumption 1 pulls the selected observation distribution back to recovered concept coordinates. The low-rank argument therefore operates on the rank-r interaction kernel S(M,P)S(c^M,c^P). If two minimal factorizations induce the same kernel, then evaluating the kernel at the full-rank protein-side probes from Condition 3.i expresses each molecule-side mode basis as a linear combination of the same finite set of kernel sections S(⋅,q)S(·,b_q). Hence the two molecule-side mode families span the same subspace. The full-rank molecule-side probes give the symmetric statement for the protein-side modes. Thus the identifiable object is the pair of interaction subspaces, not an ordered basis of individual modes. Intuitively, the rank-r subspaces are invariants of the low-rank interaction kernel, even when concept-wise support is not identifiable. This is exactly the subspace-identifiability analog we seek: we identify interaction subspaces rather than individual concept-pair entries. The full derivation is given below. A4.6.2 Intuitive Example A low-rank interaction can be dense. For example, with =[0.20.81.5],=[1.00.52.0],ℳ=⊤=[0.20.10.40.80.41.61.50.753.0], [rgb]0,0,0u= bmatrix0.2\\ 0.8\\ 1.5 bmatrix, = bmatrix1.0\\ 0.5\\ 2.0 bmatrix, =uv = bmatrix0.2&0.1&0.4\\ 0.8&0.4&1.6\\ 1.5&0.75&3.0 bmatrix, all nine concept-pair entries are nonzero, so sparse edge-level recovery is not the right target. Nevertheless, every entry is generated by one shared interaction mode: Mij=uivjM_ij=u_iv_j. A larger block mask with three dense 3×33× 3 all-ones blocks is similarly rank 33: it is dense within blocks, but generated by three interaction modes. Theorem 3 captures exactly this situation: the identifiable object is the span of the dominant modes, not the individual visual block labels or every pairwise edge. A4.6.3 Full Proof of Theorem 3 Proof. The nuisance variables do not enter this algebraic step directly: after marginalizing εM,εP ^M, ^P and using Assumption 1 to recover M,Pc^M,c^P from the observations, the remaining question is whether the rank-r interaction subspaces are uniquely determined by the kernel. Suppose two minimal rank-r factorizations induce the same interaction kernel on the data support: S(M,P)=(M)⊤(P)=~(M)⊤~(P). [rgb]0,0,0S(c^M,c^P)=u(c^M) v(c^P)= u(c^M) v(c^P). Let =spanu1,…,urU=span\u_1,…,u_r\ and ~=spanu~1,…,u~r U=span\ u_1,…, u_r\, and define ,~V, V analogously. For each protein-side probe qb_q, the kernel section S(⋅,q)=∑ℓ=1rvℓ(q)uℓ(⋅)=∑ℓ=1rv~ℓ(q)u~ℓ(⋅) [rgb]0,0,0S(·,b_q)= _ =1^rv_ (b_q)u_ (·)= _ =1^r v_ (b_q) u_ (·) belongs to both U and ~ U. Because rank(B)=rrank(V_B)=r, these sections span U. Since the same r-dimensional span is contained in ~ U and the alternative factorization is also minimal rank-r, dim(~)=r ( U)=r; hence =~U= U. The same argument with molecule-side probes ma_m and rank(A)=rrank(U_A)=r shows =~V= V. Since each pair of mode functions is a basis for the same r-dimensional subspace, there exists an invertible matrix A∈GL(r)A (r) such that, on the data support, ~=A⊤ u=A u and ~=A−1 v=A^-1v. This change of basis leaves ⊤u v unchanged but does not change the identified subspaces. Therefore the pair (M,P)(U_M,V_P) is identifiable, while the ordered mode basis is not. ∎ A4.7 Summary and Discussion of the Theory The results above establish a coherent theoretical picture for CausalBind. First, the structural causal model clarifies why binder-only training should be treated as a Heckman-style selection problem on a V-structure rather than as ordinary supervised learning. Second, Theorem 1 shows that dense holistic interaction is fundamentally non-identifiable under this selection mechanism. Third, Theorem 2 shows that sparse interaction can restore identifiability up to the standard ambiguities of permutation, component-wise reparameterization, and joint rescaling. Finally, Theorem 3 gives a subspace-identifiability result when exact sparse support is too strong: even if entry-wise mask identification fails, the dominant interaction subspace can still remain identifiable. What the three results establish together. The theoretical development above should be read as a progression from impossibility, to component-wise identifiability under strong sparse structure, to subspace identifiability under weaker but more realistic low-rank structure. Theorem 1 shows that without additional constraints, binder-only observations do not determine a unique latent interaction mechanism: many dense factorizations can explain the same selected distribution. Theorem 2 then shows that component-wise identifiability becomes possible when the interaction mask is sparse in a combinatorially distinctive way and the selected conditional family varies sufficiently across molecule-side concepts. Finally, Theorem 3 shows that even when exact element-wise sparsity is relaxed, the dominant interaction modes remain identifiable at the subspace level under a low-rank kernel assumption. Why the causal structure matters. The key source of information is the V-structure M→B←P,c^M→ B ^P, not an externally supplied domain index or an auxiliary supervision signal. Before conditioning on binding, the molecule-side and protein-side concepts are independent under Assumption 2; after restricting to B=1B=1, they become statistically coupled through the collider. The identifiability problem is therefore not solved by adding a second dataset or a separate view, but by exploiting the dependence structure already induced by binder-only selection. The sparse-support theorem uses that dependence to establish component-wise identifiability up to standard equivariances, while the low-rank theorem uses it to establish subspace identifiability of the leading interaction modes when exact support identification is too strong a requirement. Appendix A5 Implementation Details Architecture overview. For completeness, Figure A2 summarizes the implementation-level view of CausalBind. We place it in the appendix to keep the main text focused on the theory-to-method narrative, while still providing a compact diagram of the encoder, concept-factorization, sparse-interaction, and retrieval-loss pipeline. PocketMoleculeSequence3D Structure3D StructureProtein Sequencecpoc1c_poc^1cpoc2c_poc^2cpoc3c_poc^3cmol1c_mol^1cmol2c_mol^2cmol3c_mol^3cseq1c_seq^1cseq2c_seq^2cseq3c_seq^3111100111100B=1B=1 Figure A2: Three-view concept-level schematic of CausalBind. The inputs are pocket structure, molecule structure, and protein sequence. Each view is decomposed into modality-specific concepts cpocc_poc, cmolc_mol, and cseqc_seq. Local binary indicators summarize whether a concept pair is active (11) or blocked (00). We show two interaction branches, pocket-molecule and molecule-sequence, which are then aggregated into the final pair-level binding event B. Green solid arrows denote active local interactions, red dashed arrows denote blocked interactions, and gray arrows denote aggregation to the overall binding variable. A5.1 Relation Between the Two-Modal Theory and the Three-Modal Implementation The theory in Sections 2 and A4 is deliberately written for the simplest protein-molecule setting with two modalities: one molecule view and one protein view. The implementation, however, uses three modalities: molecule structure, pocket structure, and protein sequence. This does not contradict the theory; rather, it instantiates the same two-modal idea twice. Concretely, the model contains two retrieval branches, molecule-pocket and molecule-sequence, with score matrices poc,molU^poc,mol and seq,molU^seq,mol. Each branch is a protein-molecule pairing of exactly the kind analyzed in the theory, and each branch is equipped with its own constrained interaction bottleneck. The final objective simply sums the two pairwise retrieval losses and their associated regularization terms. For this reason, the main theoretical claims still apply at the level at which they were intended: they describe why dense holistic cross-modal interaction is non-identifiable, why sparse interaction can be component-wise identifiable, and why low-rank interaction can be subspace-identifiable for each protein-molecule branch individually. The three-stream architecture should therefore be read as a multi-view extension of the same pairwise principle rather than as a different causal problem. The pocket branch and the sequence branch provide two complementary protein views, but neither changes the core structural role of the protein-molecule interaction analyzed in the theory. Put differently, the implementation is richer than the theory in terms of modalities, but not different in its fundamental interaction pattern: it is still a sum of two protein-molecule pairing problems, each governed by the same sparse/low-rank inductive bias. A5.2 CausalBindSP: Sparse Concept-Pair Implementation CausalBindSP is the primary implementation used in the main text. It keeps an explicit K×K× K concept-pair mask for each protein-molecule branch and therefore most directly matches the sparse interaction form in Theorem 2. The learned dense mask can also enter the low-rank regime analyzed in Theorem 3, as diagnosed by the singular-value analyses in App. A6.1.7. Unless stated otherwise, CausalBind in the main paper refers to CausalBindSP. A5.2.1 Technical Supplement for the CausalBindSP Implementation This subsection complements Section 3 at the design level. The emphasis here is on how the abstract ingredients in the main text are instantiated in practice: how concept extraction is implemented, how the sparse mask is parameterized and applied, and how the inherited geometric losses fit into the final objective. ConceptExtractor and SparseMask. For the main sparse concept-pair operating point used in the paper, we set K=128K=128, dc=1024d_c=1024, L=4L=4 Perceiver layers, and 88 attention heads per layer; the strongest alternative concept-level settings in the sweeps vary one of these quantities at a time. The default mask type is tanh_plus_1, the hard-gating threshold is 0.50.5, and the effective mask is =tanh(ℳ^)+1,Z= ( M)+1, followed by hard thresholding so entries at or below the threshold are set to zero exactly in the forward pass. Gradients through the hard gate use a straight-through convention, and the sparsity penalty is applied to the continuous activated masks Z rather than to the raw mask logits. The main-text operating point uses λspa=10−4 _spa=10^-4, while the sweep additionally studies λ∈0,10−6,10−5,10−4,10−3,10−2,10−1,1λ∈\0,10^-6,10^-5,10^-4,10^-3,10^-2,10^-1,1\. In all cases, the branchwise penalties for the molecule-pocket and molecule-sequence masks are summed and then scaled by the chosen λspa _spa. Key implementation equations. For each branch (a,b)∈(mol,poc),(mol,seq)(a,b)∈\(mol,poc),(mol,seq)\, the sparse concept-pair implementation can be summarized as v _v =Fv(xv)∈ℝK×dc, =F_v(x_v) ^K× d_c, (31) a,bSP _a,b^SP =tanh(ℳ^a,b)+1,ℳa,bSP=HτST(a,bSP),ℳ^a,b∈ℝK×K, = ( M_a,b)+1, _a,b^SP=H_τ^ST\! (Z_a,b^SP ), M_a,b ^K× K, a,b[i,j] _a,b[i,j] =ϕ(a[i,:],b[j,:])=⟨a[i,:],b[j,:]⟩, =φ(C_a[i,:],C_b[j,:])= _a[i,:],C_b[j,:] , ua,bSP u_a,b^SP =Agg(ℳa,bSP⊙a,b). =Agg\! (M_a,b^SP _a,b ). Here FvF_v denotes the backbone-plus-concept-extractor map for modality v, τ=0.5τ=0.5 is the forward hard-gating threshold, Agg()=K−1∑i,jBijAgg(B)=K^-1 _i,jB_ij in the implementation, and ua,bSPu_a,b^SP is the branch-level score entering the contrastive objective. Detailed form of ℒconeL_cone and ℒregL_reg. In the main text we keep the HypSeek-inherited geometric terms abstract to avoid overloading Section 3. Here we make them explicit. Consider assay i with pocket embedding poc,ih_poc,i and candidate molecule embeddings mol,ijj∈ℒi\h_mol,ij\_j _i, where ℒiL_i is the set of molecules paired with assay i in the training batch. Let Nℓ=∑i|ℒi|N_ = _i|L_i| be the total number of pocket-molecule pairs. Following HypSeek [Wang et al., 2026], the Lorentz distance between pocket i and molecule j is denoted dijd_ij, the exterior angle at the pocket is ϕij _ij, and the pocket half-aperture is ωi _i. Affinity values are discretized into buckets bijb_ij, from which the bucket-specific radial and angular limits are derived as rij=r0+bijΔr,ηij=η0−bijΔη.r_ij=r_0+b_ij r, _ij= _0-b_ij η. The cone-hierarchy loss then penalizes violations in both radial distance and angular admissibility: Lrad L_rad =1Nℓ∑i,jmax(dij−rij, 0), = 1 N_ _i,j \! (d_ij-r_ij,\,0 ), (32) Lang L_ang =1Nℓ∑i,jmax(ϕij−ηijωi, 0), = 1 N_ _i,j \! ( _ij- _ij _i,\,0 ), ℒcone _cone =λradLrad+λangLang. = _radL_rad+ _angL_ang. The Nℓ N_ normalization is inherited unchanged from HypSeek; it is a code-level loss scaling convention rather than a new statistical average introduced by CausalBind. The additional regularization term in our implementation inherits two further components from HypSeek. Concretely, the ℒregL_reg appearing in the main text is exactly the weighted sum of the two terms below: ℒreg=λanguRang+λhetRhet.L_reg= _anguR_ang+ _hetR_het. The first term is an angular margin regularizer. It prevents trivial angular collapse by requiring each molecule to remain at least a fixed margin mmargin>0m_margin>0 beyond its cone boundary: Rang=1Nℓ∑i,jmax(ϕij−ηijωi+mmargin, 0).R_ang= 1 N_ _i,j \! ( _ij- _ij _i+m_margin,\,0 ). (33) Here the same ϕij _ij, ηij _ij, and ωi _i as above are reused; the only new notation is the constant angular margin mmarginm_margin, which is set to 0.150.15 in the current code. The second term is an intra-assay heterogeneity regularizer. Its role is to encourage the model to assign larger probability mass to the stronger active molecules within the same assay, rather than treating all actives as equally informative. For assay i, let Li=|ℒi|L_i=|L_i| be the number of paired molecules, and let di=(di1,…,diLi)d_i=(d_i1,…,d_iL_i) be their pocket-molecule distances. We define the within-assay rank of molecule j by rankij=1+∑j′=1Li[dij′<dij],rank_ij=1+ _j =1^L_i1[d_ij <d_ij], so smaller distance implies better rank. HypSeek then assigns a rank-based weight wij=exp(−βrankij−1Li),w_ij= \! (-β\, rank_ij-1L_i ), where β controls how strongly higher-ranked molecules are emphasized (the implementation uses β=80.5β=80.5). Let ℓij _ij denote the molecule-retrieval logit for assay i and molecule j within that assay-specific subset, and define the within-assay softmax probability pij=exp(ℓij)∑j′=1Liexp(ℓij′).p_ij= ( _ij) _j =1^L_i ( _ij ). Let vijv_ij be the activity or affinity annotation of molecule j in assay i, and let vthv_th be the HypSeek preprocessing threshold used to mark active compounds. The implementation keeps only molecules with vij<vthv_ij<v_th (equivalently, the positive subset under the assay’s activity convention). Writing C for the number of assays that contain at least one such active molecule, the heterogeneity regularizer is Rhet=1max(C,1)∑i∑j:vij<vth−wijlogpij.R_het= 1 (C,1) _i _\j:\,v_ij<v_th\-\,w_ij p_ij. (34) Intuitively, RhetR_het is a rank-weighted within-assay negative log-likelihood: stronger actives receive larger weights, so the model is encouraged to separate fine-grained affinity variation inside each pocket-specific candidate set. Relation to our Section 3 notation. In the current paper, the retrieval loss in Section 3 is written in terms of batch-level score matrices poc,molU^poc,mol and seq,molU^seq,mol. The HypSeek terms above operate on the same training pairs after they have been lifted into the hyperbolic branch-specific geometry. In other words, ℒconeL_cone and ℒregL_reg do not replace the retrieval objective; they act as geometric regularizers on top of it. This is why the main text groups them separately from ℒcontL_cont, while the appendix spells out the exact radial, angular, and within-assay ranking quantities used in the inherited implementation. A5.2.2 Concrete Parameter Assignments and Experimental Setting We now turn to the code-level side of reproducibility for CausalBindSP: what values are assigned to the key hyperparameters, how the inherited loss weights are instantiated in code, and what training/evaluation settings are shared across the reported runs. Training setup and optimization. We build on the HypSeek code base [Wang et al., 2026] with pretrained Uni-Mol encoders for molecules and pockets and an ESM-2 encoder for protein sequence. Training follows the same three-branch retrieval pipeline on the blended ChEMBL+BindingDB corpus (18,31618,316 assays after removing assays that overlap the FEP benchmarks and every assay whose target UniProt ID appears in DUD-E, LIT-PCBA, or DEKOIS 2.0; as in the LigUnity and HypSeek protocol, the PDBBind structural pairs are not filtered by UniProt ID, so some PDBBind complexes share targets with these benchmarks), with CASF used for validation when validation-selected protocols are needed. Unless noted otherwise, all reported runs train for 5050 epochs on 4×4×A100-80GB GPUs with per-GPU batch size 2424 (distributed world size 44, effective global batch size 9696), Adam optimizer, learning rate 10−410^-4, polynomial decay, warmup ratio 0.060.06, weight decay 00, gradient clipping at 1.01.0, and fp16 mixed precision. In our cluster logs and planning estimates, a typical 50-epoch run takes several hours on 4×4×A100, so full sweeps scale roughly linearly with the number of variants. The default seed is 11; additional seeds are only introduced in the dedicated robustness checks. All compared retrieval runs (HypSeek, concept-level ablations, concept-level sweeps, and embedding-level variants) share this optimization setup unless a subsection explicitly states otherwise. Inference and storage. CausalBind remains a retrieval-style dual-tower model: molecule-side and protein-side concept features are computed independently, and the fixed branch masks are applied before producing score matrices. Thus library molecules can still be precomputed rather than passed through a joint protein-molecule network for every target. The explicit K×K× K CausalBindSP masks add a small fixed branch computation, while CausalBindLR reduces mask parameters from K2K^2 to 2Kr2Kr per branch. For billion-scale deployment one would typically combine these representations with approximate-nearest-neighbor indexing or distillation; the present experiments evaluate ranking quality rather than an optimized production index. Loss weights and the relation between paper notation and code notation. In Section 3, we write the training objective using a compact notation with a sparsity weight λspa _spa and a grouped geometric term λregℒreg _regL_reg. In the code, however, the inherited HypSeek regularization is implemented through several explicit coefficients rather than through a single scalar λreg _reg. Concretely, the cone-hierarchy term uses γchl=0.1 _chl=0.1 together with λrad=0.5 _rad=0.5 and λang=0.5 _ang=0.5, while the additional regularization bundle uses λangu=0.1 _angu=0.1 and λhet=0.1 _het=0.1. Thus, the paper’s ℒregL_reg should be read as a compact umbrella notation for the inherited angular and heterogeneity regularizers, while λreg _reg is not a single independently tuned scalar in the current implementation. The only new coefficient introduced by CausalBind itself is λspa _spa, whose values are reported explicitly in the corresponding experiment tables. Evaluation protocol. We evaluate on DUD-E (102 targets) and LIT-PCBA (15 targets). For each target we rank all molecules in the test set and compute AUROC, BEDROC (α=80.5α=80.5), and EF@1%. Per-target metrics are macro-averaged across targets. Evaluation metrics. AUROC measures overall ranking quality across the full ranked list: it is the probability that a randomly chosen active is ranked above a randomly chosen inactive, so it reflects global discrimination rather than early retrieval. BEDROC80.5 is an early-enrichment metric that exponentially upweights the top of the ranked list; the parameter α=80.5α=80.5 is the standard virtual-screening setting that places roughly 80%80\% of the weight within the top ∼2% 2\% of ranks, making the metric highly sensitive to whether true actives appear near the front. EF@1% (enrichment factor at the top 1%1\%) compares the number of actives retrieved in the top 1%1\% of the ranked list against the number expected from random ranking. In practical screening, BEDROC80.5 and EF@1% are often more informative than AUROC because only a small top-ranked subset is typically advanced to experimental follow-up. A5.3 CausalBindLR: Low-Rank Concept-Basis Implementation CausalBindLR keeps the same concept extractor, retrieval branches, and losses as CausalBindSP, but replaces the unconstrained K×K× K mask logits with an explicit rank-r factorization in the concept basis. It is therefore the direct implementation of the low-rank interaction structure in Theorem 3. A5.3.1 Technical Supplement for the CausalBindLR Implementation For each branch (a,b)∈(mol,poc),(mol,seq)(a,b)∈\(mol,poc),(mol,seq)\, CausalBindLR parameterizes the raw mask as ℳ^a,b=a,ba,b⊤,a,b,a,b∈ℝK×r, M_a,b=U_a,bV_a,b , _a,b,V_a,b ^K× r, and then applies the same activation and hard gate as CausalBindSP: a,bLR=tanh(a,ba,b⊤)+1,ℳa,bLR=HτST(a,bLR).Z_a,b^LR= (U_a,bV_a,b )+1, _a,b^LR=H_τ^ST\! (Z_a,b^LR ). The raw mask logit has rank at most r and remains a concept-axis operator: it modulates the K concept tokens while preserving each token’s dcd_c-dimensional feature vector. The monotone activation and hard gate can change exact algebraic rank, but the trainable interaction directions are still controlled by the 2Kr2Kr low-rank factors. This is the key distinction from an embedding-space rank constraint, which would project pooled features into a low-dimensional feature subspace and can create a much stronger information bottleneck. Key implementation equations. Equivalently, the full CausalBindLR branch computation is ℳ^a,bLR M_a,b^LR =a,ba,b⊤=∑q=1ra,b(q)(a,b(q))⊤, =U_a,bV_a,b = _q=1^ru_a,b^(q)(v_a,b^(q)) , (35) a,bLR _a,b^LR =tanh(ℳ^a,bLR)+1, = ( M_a,b^LR)+1, ℳa,bLR _a,b^LR =HτST(a,bLR), =H_τ^ST\! (Z_a,b^LR ), a,b[i,j] _a,b[i,j] =ϕ(a[i,:],b[j,:]), =φ(C_a[i,:],C_b[j,:]), ua,bLR u_a,b^LR =Agg(ℳa,bLR⊙a,b). =Agg\! (M_a,b^LR _a,b ). Thus CausalBindLR differs from CausalBindSP only in the parameterization of the same concept-axis mask: the dense mask logits ℳ^a,b M_a,b are replaced by a rank-r factorization. The defining constraint in CausalBindLR is the rank constraint itself. The default CausalBindLR setting uses λspaLR=0 _spa^LR=0, while the optional +Lspa+L_spa diagnostic adds an L1L_1 penalty on the continuous activated low-rank masks without changing the rank-r logit factorization. Executed loss. The training loss used for the default CausalBindLR is the same code-level objective as the main implementation, with the low-rank masks substituted into the forward pass and λspaLR=0 _spa^LR=0. Concretely, the code applies the HypSeek branch retrieval loss to the molecule-pocket and molecule-sequence score matrices and then adds the inherited geometric terms; the optional +Lspa+L_spa diagnostic additionally includes the mask penalty: ℒexecLR _exec^LR =αpocℒhcc(poc,mol,LR)+αprotℒhcc(seq,mol,LR) = _poc\,L_hcc\! (U^poc,mol,LR )\;+\; _prot\,L_hcc\! (U^seq,mol,LR ) (36) +γchlℒcone+λanguRang+λhetRhet+λspaLRℒspaLR,λspaLR=0by default, + _chlL_cone+ _anguR_ang+ _hetR_het+ _spa^LRL_spa^LR, _spa^LR=0\ by default, ℒhcc() _hcc(U) =ℒpocket()+ℒmol()+ℒrank(), =L_pocket(U)+L_mol(U)+L_rank(U), ℒspaLR _spa^LR =‖molLR‖1+‖pocLR‖1+‖seqLR‖1. =\|Z_mol^LR\|_1+\|Z_poc^LR\|_1+\|Z_seq^LR\|_1. Here ℒhccL_hcc is the actual branch loss returned by compute_hcc_pair_official_style: a pocket-to-molecule term, a molecule-to-pocket term, and an intra-assay ranking term. The terms ℒconeL_cone, RangR_ang, and RhetR_het are exactly the geometric regularizers defined above for CausalBindSP. At the code level, SparseMask modules are attached to the molecule, pocket, and sequence streams; the branch-level notation in Eq. (35) is the paper-level interaction view induced by these stream masks. A5.3.2 Concrete Parameter Assignments and Experimental Setting Unless otherwise stated, CausalBindLR uses the same backbone, optimizer, training set, evaluation protocol, and concept-extractor configuration as the main CausalBindSP run: K=128K=128, dc=1024d_c=1024, L=4L=4, 88 attention heads, and threshold τ=0.5τ=0.5. The representative direct low-rank setting uses r=1r=1 and λspaLR=0 _spa^LR=0; rank scans additionally use r∈1,4,8,16,32,64r∈\1,4,8,16,32,64\. The optional +Lspa+L_spa diagnostic uses λspaLR=10−4 _spa^LR=10^-4. For K=128K=128, the rank-11 mask uses 2Kr=2562Kr=256 mask parameters per branch, compared with K2=16,384K^2=16,384 for the dense concept-pair mask. A5.4 CausalBindEMB: Embedding-Level Masking CausalBindEMB is an embedding-level alternative that applies the sparse/low-rank interaction idea after pooling. It is useful as an empirical robustness check, but it is not the primary theory-aligned method because its mask entries do not correspond to explicit concept pairs. A5.4.1 Technical Supplement for the CausalBindEMB Implementation The embedding-level variant places the learnable interaction mask after mean-pooling, rather than directly on the concept axis. Concretely, CausalBindSP first extracts K causal concepts per modality, applies the mask on the concept axis, and then pools; by contrast, CausalBindEMB first obtains pooled modality embeddings and then applies a mask ℳ∈ℝdc×dcM ^d_c× d_c in the shared embedding space. The two variants therefore share the same high-level retrieval pipeline and the same sparsity-controlled interaction idea, but differ in where the bottleneck is imposed: before pooling for CausalBindSP, versus after pooling for CausalBindEMB. This difference matters conceptually. CausalBindSP is more directly aligned with the theory in Section 2, because each entry of the mask can be interpreted as a relation between two concepts. CausalBindEMB is less interpretable in that sense, but it is architecturally simpler and still tests a related hypothesis: dense holistic interaction is unnecessary, while a constrained pooled-space interaction can already improve retrieval. Key implementation equations. The embedding-level variant uses the same concept extractor, but moves the mask to the pooled feature axis: v _v =Fv(xv)∈ℝK×dc,¯v=Pool(v)=1K∑i=1Kv[i,:], =F_v(x_v) ^K× d_c, h_v=Pool(C_v)= 1K _i=1^KC_v[i,:], (37) vEMB _v^EMB =tanh(ℳ^v)+1,ℳvEMB=HτST(vEMB),ℳ^v∈ℝdc×dc, = ( M_v)+1, _v^EMB=H_τ^ST\! (Z_v^EMB ), M_v ^d_c× d_c, ~v h_v =¯vℳvEMB,v=Pv(~v), = h_vM_v^EMB, _v=P_v( h_v), ua,bEMB u_a,b^EMB =sim(a,b). =sim(z_a,z_b). Here PvP_v is the modality-specific projection head and sim(⋅,⋅)sim(·,·) denotes the branch retrieval similarity. Unlike CausalBindSP and CausalBindLR, the mask in CausalBindEMB acts on pooled dcd_c-dimensional features, so its entries are not concept-pair interactions. Executed loss. The embedding-level implementation uses the same executed objective, but its sparsity term is applied to the pooled embedding-space masks rather than to concept-axis masks. The score matrices are formed from the embedding-level branch scores ua,bEMBu_a,b^EMB in Eq. (37): ℒexecEMB _exec^EMB =αpocℒhcc(poc,mol,EMB)+αprotℒhcc(seq,mol,EMB) = _poc\,L_hcc\! (U^poc,mol,EMB )\;+\; _prot\,L_hcc\! (U^seq,mol,EMB ) (38) +γchlℒcone+λanguRang+λhetRhet+λspaℒspaEMB, + _chlL_cone+ _anguR_ang+ _hetR_het+ _spaL_spa^EMB, ℒhcc() _hcc(U) =ℒpocket()+ℒmol()+ℒrank(), =L_pocket(U)+L_mol(U)+L_rank(U), ℒspaEMB _spa^EMB =‖molEMB‖1+‖pocEMB‖1+‖seqEMB‖1. =\|Z_mol^EMB\|_1+\|Z_poc^EMB\|_1+\|Z_seq^EMB\|_1. Thus the difference between CausalBindEMB and the concept-basis variants is not the supervision signal: all three use the same retrieval, cone, angular, and heterogeneity terms. The difference is where the sparsity-controlled mask enters the forward computation and therefore what object ℒspaL_spa regularizes. A5.4.2 Concrete Parameter Assignments and Experimental Setting The representative CausalBindEMB setting uses K=64K=64, dc=256d_c=256, L=4L=4, 88 attention heads, mask type tanh_plus_1, threshold τ=0.5τ=0.5, and λspa=10−2 _spa=10^-2. It shares the same training corpus, optimizer, GPU setup, checkpoint protocols, and evaluation metrics as CausalBindSP. In this variant the mask has size dc×dcd_c× d_c rather than K×K× K; for the representative setting, this gives 2562=65,536256^2=65,536 mask parameters per stream. A5.5 Comparison Across Three Implementations Table A3 summarizes the relationship among the three implementations. The main text prioritizes CausalBindSP because it gives the cleanest sparse concept-pair story and keeps the method section compact. CausalBindLR is the most direct architectural instantiation of the low-rank theorem, while CausalBindEMB serves as a pooled-basis alternative that is empirically useful but theoretically less direct. Table A3: Comparison of the three CausalBind implementations. The main paper presents CausalBindSP; CausalBindLR and CausalBindEMB are appendix variants. Variant Mask object Forward bottleneck Closest theory link Role in the paper CausalBindSP Dense K×K× K concept-pair mask with L1L_1 sparsity and hard thresholding. Concept axis before pooling; each active entry selects a protein-molecule concept pair. Direct sparse-support form of Theorem 2; learned masks can also enter the low-rank regime of Theorem 3. Primary method in the main text. CausalBindLR Rank-r concept-basis mask ⊤UV with ,∈ℝK×rU,V ^K× r. Concept axis before pooling; rank controls the number of interaction subspaces. Direct implementation of the low-rank subspace-identifiability structure in Theorem 3. Appendix variant for explicit low-rank implementation. CausalBindEMB Dense dc×dcd_c× d_c mask in pooled embedding space. After mean-pooling; the mask acts on pooled features rather than concept pairs. Compatible with a compressed interaction view, but not a direct instantiation of concept-pair sparse support. Appendix alternative and robustness check. The key implementation distinction is therefore the axis on which the constraint acts. CausalBindSP and CausalBindLR constrain the concept axis and preserve per-concept feature vectors, whereas CausalBindEMB constrains the pooled feature axis after the K concepts have already been averaged. This is why CausalBindSP and CausalBindLR have a cleaner theory-method correspondence, while CausalBindEMB is best read as an auxiliary empirical variant. Appendix A6 Experimental Analysis This section revisits the main-text experiments with more explanation and details. It should be read as an extension of the main experimental section rather than as a separate experimental theme. The goal here is to unpack the evidence behind the main claims, make the evaluation protocol fully explicit, and separate several empirical questions that are necessarily compressed in the main paper. The subsection layout follows the three implementations introduced in App. A5. App. A6.1 collects the results for CausalBindSP, the sparse concept-pair implementation used in the main text. App. A6.5 reports CausalBindLR, the explicit low-rank concept-basis implementation. App. A6.6 reports CausalBindEMB, the embedding-level alternative. We then synthesize the three implementations and close with the theory–practice discussion. A6.1 CausalBindSP: Sparse Concept-Pair Results A6.1.1 Extended Analyses for the Main Experiments Why does causal concept extraction help? The main paper argues that dense holistic alignment is too coarse a structural prior for protein-molecule binding, because binding is typically driven by a small number of local compatibilities rather than by uniform all-to-all agreement. The role of causal concept extraction is to operationalize this inductive bias. The concept extractor encourages each modality to decompose into a smaller set of binding-relevant factors, and the sparse interaction mask then decides which cross-modal pairs matter for prediction. In the scratch-training setting, this provides a strong bias toward disentangled structure. In the pretrained setting considered in our main experiments, it plays a more selective role: rather than learning chemistry and protein semantics from nothing, the module filters a pretrained representation and suppresses features that are predictive but not genuinely binding-relevant. Why do the gains concentrate on early enrichment? Across both experimental settings, the improvements in BEDROC80.5 and EF@1% are often larger than the improvements in AUROC. This pattern is important rather than incidental. AUROC measures global ranking quality over the entire candidate list, whereas BEDROC80.5 and EF@1% emphasize whether true actives are pushed into the very top of the ranked list, which is the practically important regime in virtual screening. Our interpretation is that the sparse interaction prior is especially helpful when the model must decide which few candidates deserve the highest confidence, rather than when it is merely asked to produce a smooth global ordering. Why is LIT-PCBA the stricter test? The larger improvements on LIT-PCBA than on DUD-E suggest that the causal prior is most useful when shortcut features become unreliable. DUD-E remains a useful benchmark because of its scale and broad target coverage, but LIT-PCBA is deliberately constructed to reduce property-based biases and therefore makes it harder for a model to succeed by exploiting easy decoy artifacts. When improvements persist on LIT-PCBA, especially in early-enrichment metrics, they are more plausibly reflecting better binding discrimination rather than better exploitation of dataset quirks. This is why the appendix repeatedly treats LIT-PCBA as the main stress test for whether the sparse interaction prior is doing the intended job. A6.1.2 Main Comparison: Three-Protocol Evaluation The main text reports the last checkpoint for all methods because, in this cross-dataset generalization setting, it is the cleanest fixed-endpoint protocol and avoids tuning checkpoint selection to the test benchmark. Here we additionally report two validation-selected alternatives, best_auc and best_bedroc, to show how much the main conclusions depend on checkpoint choice. To keep the comparison unambiguous, Table A4 uses a single seed throughout and does not mix in additional exploratory variants. The three methods shown are the two strongest retrieval baselines under our pipeline, HypSeek and LigUnity, together with our main concept-level configuration (K=128K=128, dc=1024d_c=1024, λ=10−4λ=10^-4, L=4L=4) and its atom-attentive version CausalBindATOM under the same configuration. Table A4: Three-protocol evaluation for main comparisons. The main text reports the fixed-endpoint last protocol; best_auc and best_bedroc are complementary validation-selected references. All rows are seed-1 results. Method Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% LigUnity last 0.897 0.674 44.20 0.599 0.075 6.50 best_auc 0.915 0.716 46.42 0.581 0.089 8.15 best_bedroc 0.896 0.671 44.00 0.599 0.076 6.38 HypSeek last 0.909 0.605 37.83 0.603 0.061 4.62 best_auc 0.913 0.629 39.63 0.608 0.063 5.13 best_bedroc 0.923 0.676 42.71 0.607 0.063 4.56 CausalBindSP last 0.935 0.744 47.69 0.639 0.097 8.56 best_auc 0.934 0.722 46.41 0.628 0.074 5.84 best_bedroc 0.935 0.747 48.24 0.643 0.094 8.42 CausalBindATOM last 0.943 0.768 49.66 0.621 0.079 6.25 best_auc 0.949 0.749 47.51 0.623 0.070 6.02 best_bedroc 0.943 0.768 49.55 0.621 0.084 6.75 Two points matter for interpretation. First, the protocol choice does not reverse the main qualitative ranking between HypSeek and CausalBindSP: under all reported checkpoints, our concept-level model remains clearly stronger on LIT-PCBA, which is the more realistic benchmark in our evaluation. The largest gap appears under the fixed-endpoint last protocol used in the main text, where LIT-PCBA BEDROC80.5 improves from 0.0610.061 to 0.0970.097 and EF@1% improves from 4.624.62 to 8.568.56. Second, the three protocols emphasize different aspects of the same model. For CausalBindSP, best_bedroc slightly improves DUD-E and raises LIT-PCBA AUROC to 0.6430.643, but last gives the strongest LIT-PCBA early-enrichment profile, which is precisely the regime we care about in practical screening. The best_auc checkpoint is noticeably weaker on LIT-PCBA for our model, showing that checkpoint selection by a validation AUC criterion does not necessarily align with the practical objective of top-ranked hit discovery. LigUnity shows a useful contrast. On DUD-E, validation-selected checkpoints improve over its last endpoint, especially under best_auc. On LIT-PCBA, however, the picture is mixed: best_auc improves early enrichment over its last result but lowers AUROC, while best_bedroc stays close to the fixed-endpoint baseline. This reinforces the same lesson as for CausalBindSP: checkpoint selection changes the metric emphasis, but it does not overturn the main comparison. Across all three protocols, CausalBindSP remains stronger than LigUnity on the LIT-PCBA metrics emphasized in the main text. CausalBindATOM gives the strongest DUD-E results under every protocol, whereas its LIT-PCBA early enrichment stays below that of CausalBindSP and varies with checkpoint choice (6.026.02–6.756.75 EF@1%), mirroring the pattern of the main comparison. A6.1.3 Ablation and Diagnostic Study: Detailed Analysis The ablation and diagnostic table in the main text is intended to answer a mechanistic question rather than to produce a second leaderboard: which structural biases are responsible for the improvement, and how should we interpret the fact that different constraints matter differently across benchmarks? This distinction is important because a component ablation need not improve every metric monotonically in order to be scientifically informative. In particular, the LIT-PCBA rows should be read as evidence about which inductive bias is doing what, not as evidence that each added module must dominate under every protocol and every metric. Table 2 should be read in two parts: the CausalBindSP rows are direct component ablations, whereas the CausalBindLR and CausalBindEMB rows are diagnostics for where and how rank compression is imposed. Reading the rows from top to bottom reveals the following: (1) HypSeek and LigUnity establish the dense-alignment baseline. These two rows show what current retrieval systems can do without the explicit concept-level interaction structure introduced in CausalBind. They provide the reference point for asking whether constrained concept-level interaction improves over dense holistic retrieval. (2) The full model (CausalBindSP) gives the strongest overall balance. This row is the best summary of our main claim. On DUD-E it is the clear winner on all three metrics. On LIT-PCBA it gives the best AUROC and remains strong on BEDROC80.5 and EF@1%, so the full architecture gives the best cross-benchmark tradeoff. (3) Removing only LspaL_spa isolates an unguided learned mask. The cross-modal mask is still present, but it is no longer coupled to an explicit sparsity objective. The resulting drop suggests that the mask is most useful when regularized toward a simple interaction structure. (4) The LR and EMB diagnostics separate two kinds of compression. In CausalBindLR, both rows keep the rank-one concept-axis factorization; adding the optional LspaL_spa penalty changes little, indicating that the low-rank parameterization itself is the operative constraint. In CausalBindEMB, however, the embr=1M_emb^r=1 row imposes rank one after pooling and therefore becomes a much stronger information bottleneck. Main takeaway from the ablation. The cleanest reading is that the constrained concept-level contrastive route is the essential structural change. The concept extractor and mask help because they regularize how contrastive learning sees the two modalities, and the comparison with CausalBindLR shows that explicit low-rank concept-axis compression can provide a similar structural bias. Connection to the later theory–practice analysis. This reading is consistent with the mask-structure results. The useful empirical regime is not aggressively hard-sparse, but moderately regularized and low-rank. The ablation therefore supports the same calibrated view as the theory–practice analysis: structural constraints matter, but their benefit comes from shaping the interaction into a simple reusable form rather than from adding parameters. A6.1.4 Hyperparameter Study on K and dcd_c The next question is whether the selected concept count K and concept dimension dcd_c are robust design choices or just narrow local optima. We therefore report the full protocol-resolved scans for both quantities. These tables are more informative than the single operating-point curves in the main text because they make two things visible at once: where performance is consistently strong, and where apparent improvements depend strongly on the checkpoint-selection rule. Table A5: CausalBindSP concept-level K-scan (fix dc=512d_c=512), seed=1, all three protocols. K=256K=256 underperforms consistently. K Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 32 last 0.938 0.749 48.41 0.620 0.081 6.90 best_auc 0.939 0.750 48.38 0.625 0.083 6.85 best_bedroc 0.938 0.750 48.57 0.621 0.081 6.90 64 last 0.930 0.724 46.55 0.637 0.085 6.41 best_auc 0.933 0.715 45.90 0.638 0.088 7.88 best_bedroc 0.935 0.732 47.08 0.635 0.079 5.96 128 last 0.943 0.752 47.93 0.633 0.081 7.06 best_auc 0.943 0.755 48.33 0.634 0.082 7.04 best_bedroc 0.943 0.752 47.93 0.632 0.082 7.06 256 last 0.916 0.660 41.50 0.610 0.066 5.90 best_auc 0.917 0.662 41.82 0.609 0.065 5.46 best_bedroc 0.916 0.664 42.09 0.612 0.069 6.02 Table A6: CausalBindSP concept-level dcd_c-scan (fix K=128K=128), seed=1, all three protocols. dc=1024d_c=1024 gives the strongest LIT-PCBA results. dcd_c Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 128 last 0.919 0.589 35.75 0.615 0.062 5.25 best_auc 0.922 0.629 39.43 0.630 0.060 4.75 best_bedroc 0.920 0.592 36.03 0.620 0.062 5.35 256 last 0.912 0.600 36.93 0.614 0.062 5.68 best_auc 0.922 0.683 43.07 0.620 0.067 4.73 best_bedroc 0.918 0.642 40.05 0.621 0.070 6.09 512 last 0.943 0.752 47.93 0.633 0.081 7.06 best_auc 0.943 0.755 48.33 0.634 0.082 7.04 best_bedroc 0.943 0.752 47.93 0.632 0.082 7.06 1024 last 0.935 0.744 47.69 0.639 0.097 8.56 best_auc 0.934 0.722 46.41 0.628 0.074 5.84 best_bedroc 0.935 0.747 48.24 0.643 0.094 8.42 Interpretation. The concept-level sweep shows a broad stable region rather than a single fragile optimum. In Table A5, K=128K=128 is the most consistent choice across all three checkpoint protocols, while K=256K=256 degrades on both DUD-E and LIT-PCBA. In Table A6, increasing dcd_c from 512512 to 10241024 trades a small amount of DUD-E margin for a clear gain on LIT-PCBA early enrichment, which is why the main text uses the higher-capacity setting. A6.1.5 Hyperparameter Study on λ This sweep asks a more specific question than the K/dcK/d_c tables above: once the concept budget is fixed at the main operating point (K=128K=128, dc=1024d_c=1024, L=4L=4), what range of sparsity-loss weights produces useful behavior, and where does the model collapse? We therefore vary the L1L_1 sparsity weight λ across eight orders of magnitude and report all three checkpoint protocols. The resulting pattern is useful for interpreting both the main-text hyperparameter plot and the theory–practice discussion that follows in Section A6.1.7. Table A7: CausalBindSP concept-level λ-scan, seed=1, all three protocols. K=128K=128, dc=1024d_c=1024, L=4L=4. Rows below the dotted rule show mask collapse under last and best_bedroc. λ Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 00 last 0.938 0.746 48.14 0.617 0.080 6.83 best_auc 0.943 0.744 47.68 0.629 0.073 6.26 best_bedroc 0.934 0.732 47.03 0.613 0.075 6.10 10−610^-6 last 0.940 0.749 47.94 0.635 0.088 7.26 best_auc 0.941 0.750 47.75 0.628 0.086 7.41 best_bedroc 0.941 0.755 48.38 0.631 0.086 7.67 10−510^-5 last 0.938 0.735 47.09 0.607 0.078 6.29 best_auc 0.940 0.732 47.08 0.628 0.082 6.41 best_bedroc 0.940 0.735 47.15 0.613 0.070 5.72 −10^-4 last 0.935 0.744 47.69 0.639 0.097 8.56 best_auc 0.934 0.722 46.41 0.628 0.074 5.84 best_bedroc 0.935 0.747 48.24 0.643 0.094 8.42 10−310^-3 last 0.943 0.768 49.73 0.614 0.073 5.49 best_auc 0.936 0.712 45.34 0.620 0.072 5.89 best_bedroc 0.936 0.712 45.34 0.620 0.072 5.89 —- mask collapses below (last and best_bedroc protocols) —- 10−210^-2 last 0.437 0.013 00.53 0.457 0.011 0.46 best_auc 0.921 0.699 44.71 0.622 0.074 5.83 best_bedroc 0.442 0.014 00.64 0.460 0.011 0.54 10−110^-1 last 0.297 0.006 00.30 0.437 0.013 0.59 best_auc 0.930 0.696 44.53 0.619 0.078 6.73 best_bedroc 0.930 0.696 44.53 0.619 0.078 6.73 11 last 0.725 0.099 05.30 0.595 0.028 1.44 best_auc 0.936 0.721 46.22 0.642 0.072 5.87 best_bedroc 0.930 0.697 43.99 0.635 0.064 4.92 Interpretation. Two regimes are practically useful. λ=10−4λ=10^-4 and λ=10−3λ=10^-3 both give strong results, but they emphasize different benchmarks: the former is clearly better on LIT-PCBA, while the latter is strongest on DUD-E. By contrast, λ≤10−6λ≤10^-6 is almost indistinguishable from turning the sparsity loss off, and λ≥10−2λ≥10^-2 pushes the hard gate into collapse under the fixed-endpoint and BEDROC-selected protocols. The best_auc rows remain more resilient in that high-λ region because they often pick an earlier checkpoint, before the mask has fully saturated. A6.1.6 Hyperparameter Study on L This scan isolates a different modeling choice: once the concept count, channel width, and sparsity weight are fixed, how much depth is actually useful in the concept extractor itself? Since the main text argues that the key inductive bias comes from structured cross-modal interaction rather than depth for its own sake, this table is mainly a check that deeper extractors do not hide an unreported gain. Table A8: CausalBindSP concept-level Perceiver-depth (L) scan, seed=1, all three protocols. K=128K=128, dc=1024d_c=1024, λ=10−4λ=10^-4. L Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 2 last 0.940 0.763 49.20 0.625 0.087 6.98 best_auc 0.946 0.776 49.99 0.616 0.082 6.53 best_bedroc 0.942 0.757 48.48 0.622 0.085 7.95 4 (default) last 0.935 0.744 47.69 0.639 0.097 8.56 best_auc 0.934 0.722 46.41 0.628 0.074 5.84 best_bedroc 0.935 0.747 48.24 0.643 0.094 8.42 66 last 0.936 0.738 47.05 0.633 0.085 6.75 best_auc 0.937 0.729 46.70 0.634 0.086 6.89 best_bedroc 0.937 0.733 46.79 0.629 0.086 6.93 88 last 0.934 0.727 46.38 0.630 0.080 6.73 best_auc 0.938 0.724 46.32 0.622 0.072 6.05 best_bedroc 0.931 0.702 45.03 0.638 0.079 6.38 Interpretation. The depth sweep is simple: modest depth is enough. L=2L=2 is strongest on DUD-E under best_auc, while L=4L=4 gives the best LIT-PCBA early enrichment under the two protocols that matter most for our main narrative, last and best_bedroc. Increasing depth further to L=6L=6 or 88 never helps. This is consistent with the paper’s central claim that the structural bias comes mainly from the concept factorization and cross-modal mask, not from stacking a deeper extractor. A6.1.7 Detailed Mask Structural Analysis The compact λ sweep in Table 3 motivates a more detailed question: does the learned interaction become element-wise sparse, spectrally low-rank, or both? We therefore measure both element-wise sparsity and spectral low-rank structure directly on the learned concept-level masks. Table A9 reports code-level stream diagnostics for the molecule, pocket, and sequence mask modules (K=128K=128), while the main-text Table 3 reports compact paired-branch statistics for the two protein-molecule retrieval branches. In the notation of Section 2, Condition 2.i corresponds to element-wise sparse support, whereas Condition 3.i corresponds to low-rank or mode-compressed interaction. Measurement. Let ℳeff=σ(ℳ)=tanh(ℳraw)+1∈(0,2)M_eff=σ(M)= (M_raw)+1∈(0,2). The concept-level forward pass applies a hard gate: entries with ℳeff≤0.5M_eff≤ 0.5 are zeroed exactly. The L1L_1 mass and spectral ranks below are measured on the continuous pre-gate ℳeffM_eff, while the zero counts are measured after the forward threshold. We report: • Element-wise sparsity (corresponds to Condition 2.i): fraction of entries with ℳeff<0.5M_eff<0.5, i.e. the number of hard zeros after the forward gate. • Spectral effective rank (corresponds to Condition 3.i): the smallest r such that the top-r singular values explain at least 95%95\% or 99%99\% of ‖ℳeff‖F2\|M_eff\|_F^2. • Summary scalars: the average mask value σ(ℳ)¯ σ(M) and the total mask mass L1=∑ijσ(ℳ)ijL_1= _ijσ(M)_ij. • Antichain violations (corresponds to the antichain clause of Condition 2.i): on the hard support [ℳeff>0.5]1[M_eff>0.5] of each cross-modal mask (molecule–pocket and molecule–sequence), we compare every pair of distinct row supports and every pair of distinct column supports, and count a violation whenever one support is contained in the other (equal supports also count). Each branch contributes (1282)=8,128 1282=8,128 row pairs and 8,1288,128 column pairs, giving 32,51232,512 tests in total. This statistic is reported in the last column of Table 3. The antichain statistic sharpens the two-regime interpretation. With no hard zeros (λ≤10−4λ≤ 10^-4), all row and column supports equal the full index set, so every pair violates the antichain condition; these operating points should be interpreted only through Condition 3.i and Theorem 3. At λ=10−3λ=10^-3, every one of the 128128 row and 128128 column supports is unique in both branches, with zero containment relations, so the antichain clause is exactly satisfied by the learned hard support. At λ=10−2λ=10^-2 the masks remain exact antichains but retrieval collapses, so the antichain property alone does not explain performance. We note that the λ=10−3λ=10^-3 mask remains only moderately sparse (79.5%79.5\% active entries) and that Condition 2.i is not directly measured; the diagnostic validates the antichain subcondition rather than every premise of Theorem 2. All statistics are computed exactly on the same single-seed last checkpoints as Table 3. This subsection uses the per-stream quantities σ(ℳ)¯ σ(M) and L1=∑ijσ(ℳ)ijL_1= _ijσ(M)_ij because Table A9 reports stream-level mask structure. By contrast, the main-text Table 3 reports the aggregated statistic ‘L1L_1 avg’, i.e., the average absolute mask entry after averaging over all entries and then over the two paired retrieval branches. The two views are consistent but not the same object: one is a stream-level total-mass diagnostic, while the other is a normalized paired-branch summary. Operational criteria for “sparse” and “low-rank”. The theory itself does not specify a single numerical threshold such as “10% zeros” or “rank below 5”; the conditions are structural. Still, for the purpose of interpreting the experiments, it is useful to make the comparison quantitative. Here K=128K=128, so Condition 3.i and the low-rank regime of Theorem 3 are most plausible when the effective rank needed to explain the mask energy is much smaller than 128128. By that standard, the evidence for low-rank structure is strong: for λ=10−4λ=10^-4 we have r@95%=1r@95\%=1 on all three stream masks and r@99%∈9,12,14r@99\%∈\9,12,14\, i.e., only about 7%7\%–11%11\% of the full concept dimension is needed to explain 99%99\% of the squared singular-value energy. For λ≥10−3λ≥10^-3 the evidence is even stronger in rank terms, since both r@95%r@95\% and r@99%r@99\% collapse to 11. For Condition 2.i and the sparse-support regime of Theorem 2, the practically relevant operating point is λ=10−3λ=10^-3. Condition 2.i requires a genuinely sparse support pattern together with the antichain-style separation discussed in the theory. At λ=10−3λ=10^-3, the three stream masks show about 9.7%9.7\%–13.4%13.4\% exact zeros while still maintaining strong retrieval performance, which is exactly the qualitative behavior expected from the sparse-support theorem. At λ=10−4λ=10^-4, by contrast, there are no hard zeros at all, so this setting should be interpreted through the low-rank theorem rather than the sparse one. In this sense, the experiments support two different theorem-aligned regimes: λ=10−3λ=10^-3 for sparse-support identifiability and λ=10−4λ=10^-4 for low-rank subspace identifiability. Table A9: Mask structural analysis across λ for the concept-level model. Per-stream numbers are reported at K=128K=128 under the last protocol, seed=1. L1L_1 and rank statistics use the continuous pre-gate masks; zeros use the hard-gated forward masks. λ=10−4λ=10^-4 remains element-wise dense but has a richer spectral tail, while λ≥10−3λ≥10^-3 produces hard zeros and rank-1 masks. λ branch σ(ℳ)¯ σ(M) max σ(ℳ)σ(M) L1L_1 (sum) zeros %zeros r@95%r@95\% r@99%r@99\% 00 mol 0.920 1.311 15068 0000 00.0% 1 5 poc 0.993 1.398 16267 0000 00.0% 1 1 seq 1.023 1.374 16758 0000 00.0% 1 1 10−610^-6 mol 0.910 1.301 14914 0000 00.0% 1 5 poc 0.991 1.395 16228 0000 00.0% 1 1 seq 1.023 1.374 16755 0000 00.0% 1 1 10−510^-5 mol 0.894 1.286 14648 0000 00.0% 1 6 poc 0.968 1.377 15863 0000 00.0% 1 2 seq 1.007 1.361 16505 0000 00.0% 1 1 −10^-4 mol 0.756 1.148 12383 0000 00.0% 1 14 poc 0.781 1.200 12793 0000 00.0% 1 12 seq 0.841 1.206 13774 0000 00.0% 1 9 −10^-3 mol 0.512 0.773 08383 01645 10.0% 1 1 poc 0.512 0.797 08390 02198 13.4% 1 1 seq 0.513 0.751 08404 01593 09.7% 1 1 10−210^-2 mol 0.501 0.656 08206 08677 53.0% 1 1 poc 0.502 0.708 08222 08448 51.6% 1 1 seq 0.502 0.659 08222 09199 56.1% 1 1 10−110^-1 mol 0.500 0.599 08195 11796 72.0% 1 1 poc 0.500 0.653 08199 11484 70.1% 1 1 seq 0.500 0.610 08199 11790 72.0% 1 1 11 mol 0.500 0.571 08193 13233 80.8% 1 1 poc 0.500 0.613 08194 12994 79.3% 1 1 seq 0.500 0.561 08194 13075 79.8% 1 1 Findings. Three patterns matter most. First, the effective masks are low-rank throughout the sweep: r@95%r@95\% is always 11, so the dominant interaction mode is already highly compressed even without explicit sparsity. Second, λ=10−4λ=10^-4 does not create hard zeros, but it does increase the spectral tail to r@99%=9r@99\%=9–1414, which is exactly the regime where LIT-PCBA performance is strongest. Third, once λ reaches 10−310^-3 and above, hard zeros appear abruptly and the masks collapse back to rank-11 even at the 99% level. In other words, stronger sparsity regularization does not gradually sharpen a useful structure; it eventually removes the richer low-rank structure that seems to matter most in practice. How the conditions appear in practice. The current measurements support both theorem regimes, but through different diagnostics. Condition 3.i is fundamentally about dimensionality reduction of the interaction kernel, and singular-value diagnostics measure exactly that, so λ=10−4λ=10^-4 gives direct quantitative support for the low-rank theorem. Condition 2.i is about sparse support structure, and the hard-zero statistics at λ=10−3λ=10^-3 show that the model has indeed entered that sparse-support regime while retaining strong retrieval quality. The table therefore supports a practical two-regime reading rather than a single-regime one. Best-effort conclusion about the assumptions. The most useful reading is therefore complementary rather than asymmetric. For λ=10−3λ=10^-3, the experiments support the sparse-support theorem in the regime where meaningful exact zeros appear and performance remains strong. For λ=10−4λ=10^-4, the experiments support the low-rank theorem in the regime where the mask remains dense entry-wise but is strongly spectrally compressed. The two theorems are therefore both realized in practice, but at different operating points of the same model. Implication for theory. The correspondence to the theory is therefore not “Theorem 2 fails and only Theorem 3 matters.” It is that the model exhibits two neighboring theorem-aligned regimes. Around λ=10−3λ=10^-3, the learned mask behaves in the sparse-support sense emphasized by Theorem 2. Around λ=10−4λ=10^-4, it behaves in the low-rank sense emphasized by Theorem 3. Since both settings remain empirically strong, the theory is useful in exactly the way we want: it explains two distinct but valid operating points of the learned interaction. A6.1.8 ROC-Enrichment (RE) on DUD-E ROC-enrichment (RE) is another early-recognition metric, complementary to BEDROC80.5 and EF@1%. At a fixed false-positive-rate threshold α, it is defined as RE@α=TPR(α)/αRE@α=TPR(α)/α, so larger values mean that the model retrieves actives much more efficiently than random ranking in the very early part of the list. Following prior reporting practice [Wang et al., 2026], we report RE at 0.5%0.5\%, 1%1\%, 2%2\%, and 5%5\% false-positive-rate thresholds on DUD-E. We report RE only on DUD-E because it is the more standard setting in prior work and provides a more stable target-averaged estimate than LIT-PCBA. Concretely, DUD-E contains 102102 targets, whereas LIT-PCBA contains only 1515. Table A10 therefore augments the main-text DUD-E comparison with RE, while keeping AUROC / BEDROC80.5 / EF@1% for context. To stay aligned with the main text, all rows use the same fixed-endpoint last checkpoint protocol after the full training budget. Table A10: ROC-enrichment (RE) on DUD-E under the last checkpoint protocol. Values are 102102-target means. AUROC / BEDROC80.5 / EF@1% are reproduced from the main DUD-E comparison for context. Configuration Primary metrics ROC-enrichment (RE) AUROC BEDROC80.5 EF@1% @0.5% @1% @2% @5% DrugCLIP [Gao et al., 2023] 0.809 0.505 31.89 73.97 41.79 23.68 11.16 LigUnity [Feng et al., 2025] 0.897 0.674 44.20 104.69 57.47 33.76 13.88 HypSeek [Wang et al., 2026] 0.909 0.605 37.83 95.20 54.49 30.74 14.01 CausalBindSP (K=128,dc=512,L=4K=128,d_c=512,L=4) 0.943 0.752 47.93 127.56 69.94 37.83 16.39 CausalBindSP (K=128,dc=1024,L=4K=128,d_c=1024,L=4) 0.935 0.744 47.69 128.06 69.26 37.07 15.99 CausalBindSP (K=128,dc=1024,L=2K=128,d_c=1024,L=2) 0.940 0.763 49.20 131.55 71.17 38.04 16.30 Detailed analysis. Table A10 supports three conclusions. First, CausalBindSP improves very-early retrieval not only relative to the HypSeek reproduction, but also relative to the broader retrieval baselines included in the main comparison. All three reported CausalBindSP settings are clearly stronger than DrugCLIP and LigUnity on AUROC, BEDROC80.5, EF@1%, and all four RE thresholds. This is useful because RE asks a slightly different question from BEDROC80.5 and EF@1%: instead of weighting the top of the ranking or counting actives in the top percentile, it measures how efficiently true actives are recovered at extremely small false-positive rates. The consistent improvement across all of these views strengthens the claim that CausalBindSP genuinely improves early retrieval on DUD-E. How to read the CausalBindSP rows. Second, the ordering among the three CausalBindSP settings is informative. The main-text operating point (K=128K=128, dc=1024d_c=1024, L=4L=4) is already much stronger than the prior baselines, but the shallower L=2L=2 variant is slightly better on the RE thresholds and on DUD-E BEDROC80.5 / EF@1%. This matches the broader hyperparameter-study pattern that, on DUD-E alone, a slightly shallower interaction depth can be favorable for very-early retrieval. Third, this does not overturn the model choice in the main text. The paper uses L=4L=4 because it gives the better cross-benchmark balance once LIT-PCBA is taken into account, whereas RE is reported only on DUD-E as a supplementary diagnostic. We therefore read Table A10 as supporting evidence for the DUD-E early-recognition story, not as a new primary criterion for model selection. A6.1.9 Affinity Ranking on Merck-FEP-8 We additionally evaluate whether the main CausalBindSP checkpoint transfers from virtual screening to affinity ranking. We use HypSeek’s test_fep() pipeline on the 16 FEP benchmark targets, consisting of JACS-8 and Merck-FEP-8; because the current comparison focuses on structurally diverse affinity-ranking targets, Table A11 reports only Merck-FEP-8. The evaluation uses the single-seed last checkpoint protocol for CausalBindSP and the reproduced HypSeek baseline. Missing bench/ CSV files only affect ACC/AUC outputs, which are skipped; Pearson r and Spearman ρ are unaffected. Table A11: Merck-FEP-8 affinity-ranking correlations. Higher Pearson r and Spearman ρ are better. CausalBindSP uses the main K=128,dc=1024K=128,d_c=1024 configuration under seed 1 and the last checkpoint. Method Pearson r Spearman ρ LigUnity 0.532 0.499 HypSeek 0.423 0.390 CausalBindSP 0.598 0.548 On Merck-FEP-8, CausalBindSP improves over HypSeek by +0.175+0.175 Pearson r and +0.158+0.158 Spearman ρ, and also exceeds LigUnity by +0.066+0.066 and +0.049+0.049, respectively. This suggests that the sparse concept-pair interaction learned for retrieval also transfers to affinity ranking on diverse ligand sets. A6.2 Comparison with S2Drug S2Drug [He et al., 2026] is a concurrent retrieval-based virtual-screening method. Because its implementation was not publicly available, we could not reproduce it under our pipeline; Table 1 therefore reports its paper-reported results, and this comparison should be read as paper-to-paper rather than controlled. S2Drug reports EF@0.5% as its primary LIT-PCBA enrichment metric, whereas our main table follows HypSeek and reports EF@1%. Table A12 provides the matched EF@0.5% comparison, computed by the same evaluator from the same checkpoints as Table 1. At the matched cutoff, CausalBindLR and CausalBindEMB exceed the paper-reported S2Drug result, while CausalBindSP is comparable; at EF@1%, all three variants exceed it. Table A12: LIT-PCBA EF@0.5% comparison with the paper-reported S2Drug result. Method EF@0.5% EF@1% S2Drug (paper-reported) 11.44 7.38 CausalBindSP 10.88 8.56 CausalBindLR 11.59 8.84 CausalBindEMB 12.20 9.60 A6.3 Out-of-Distribution Evaluation on DEKOIS 2.0 To test generalization beyond DUD-E and LIT-PCBA, we perform zero-shot evaluation on the established DEKOIS 2.0 benchmark [Bauer et al., 2013] under two complementary settings, both fixed before running any model. Target-level OOD retains the nine DEKOIS targets whose UniProt IDs are absent from all training sources, including PDBBind (11BETAHSD1, ACE2, ADRB2, ALR2, COX1, CYP2A6, INHA, MMP2, and TK). Scaffold-level OOD further retains only test pairs whose Bemis–Murcko scaffolds are absent from all 439,639439,639 unique training molecules; requiring at least five such actives per target leaves six targets, 5,3855,385 pairs, and 9898 actives. Metrics are macro-averaged over targets. Table A13: Zero-shot out-of-distribution evaluation on DEKOIS 2.0. CausalBindATOM is the atom-attentive version of CausalBindSP (Section 4). Method Target-level OOD (9 targets) Scaffold-level OOD (6 targets) AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% HypSeek [Wang et al., 2026] 0.819 0.593 20.06 0.880 0.630 44.47 LigUnity [Feng et al., 2025] 0.859 0.570 19.82 0.915 0.612 43.11 CausalBindSP 0.860 0.658 23.00 0.945 0.712 43.66 CausalBindLR 0.867 0.653 22.71 0.949 0.716 44.70 CausalBindATOM 0.837 0.666 22.94 0.941 0.744 50.77 Table A13 shows that the structured-interaction models improve AUROC and BEDROC80.5 over both retrieval baselines under both shifts, while CausalBindSP remains comparable to HypSeek at the strictest EF@1% cutoff of the scaffold-level split. Together with the cross-dataset protocol of the main experiments, these results indicate that the learned interaction structures transfer under explicit target-level and scaffold-level distribution shifts. Under CD-HIT sequence clustering at 40% identity, only one of the nine targets (ACE2) has its entire cluster absent from training, so these results should not be read as evidence of broad protein-family generalization. A6.4 Interpretability Case Study on THRB To connect the learned concepts with biologically meaningful binding patterns, we analyze the aligned protein–ligand crystal complex of the Human Thyroid Hormone Receptor β (THRB) from DUD-E, following the evidence chain from local structural contacts to learned concepts, to the predicted binding score, and finally to a random-control validation (Table A14, Figure A3). Observed ligand–pocket contacts are heavy-atom pairs within 4 Å. Concepts are selected solely by their attention coverage of these contacts, using mean-over-heads attention from the final Perceiver layer, without using the downstream score change. Among the contacting atoms, the four ligand atoms and eight pocket atoms most attended by the selected concepts are then excluded from the concept bottleneck while the Uni-Mol encoder outputs are kept fixed, and the final pocket–ligand score is recomputed. As a control, we repeat this procedure for 64 matched random sets of four ligand and eight pocket atoms. Table A14: Local-to-global attribution on the DUD-E THRB complex. Stage Evidence Observation Purpose Local structure Heavy-atom contacts (≤4≤ 4 Å) 98 contacts involving 27 ligand and 47 pocket atoms Identify the local binding interface Learned concept Contact concentration in the most contact-aligned concept pair 0.0505 vs. 0.00881 uniform baseline (5.73× enrichment) Test whether contacts concentrate in concepts Predicted score Mask the associated 4 ligand and 8 pocket atoms (encoder fixed) Score decreases 1.878→0.8511.878→ 0.851 (−54.7%-54.7\%) Test contribution to the prediction Random control Mask 64 matched random sets of 4 ligand and 8 pocket atoms Mean decrease 0.026±0.0730.026± 0.073; guided decrease 1.0271.027 (99.2nd percentile) Exclude an arbitrary masking effect Figure A3: Local-to-global attribution on the DUD-E THRB complex. (A) Ligand–pocket heavy-atom contacts within 4 Å in a 2D projection; highlighted atoms are those occluded at the concept bottleneck. (B) Attention-weighted contact probability over ligand–pocket concept pairs, with the contact-selected concepts and the top pair marked. (C) Decrease in the final binding score after masking the local contact atoms, matched random atoms, or the contact-selected concepts. The highest-contributing contacts are chemically plausible close polar contacts: ligand N–Asp219 OD1 (2.71 Å), N–Asp219 OD2 (2.78 Å), N–Gly250 O (2.88 Å), and O–Gly248 N (3.04 Å). These results show that biologically meaningful local contacts concentrate within a learned concept pair, and that concept-guided masking causes a much larger score decrease than random masking, supporting the intended local-structure-to-concept-to-score mechanism rather than an arbitrary masking effect. This post-hoc diagnostic uses CausalBindATOM, which exposes the full Uni-Mol atom sequence to the Perceiver; it does not claim that individual concepts carry one-to-one biochemical meanings. Attention and occlusion provide model-level attribution evidence rather than proof of a physical causal mechanism. A6.5 CausalBindLR: Low-Rank Concept-Basis Results This subsection reports the explicit low-rank concept-basis implementation, denoted CausalBindLR in the paper and recorded as V3 in the experiment log. It uses the same concept extractor and retrieval loss as CausalBindSP, but replaces the dense K×K×K concept-pair mask with a hard rank-r factorization. Thus it is the most direct empirical test of the subspace-identifiability view in Theorem 3. A6.5.1 Main Rank Scan and Three-Protocol Results For each branch, CausalBindLR parameterizes the concept-basis mask as σ(ℳLR)=tanh(⊤)+1,ℳeff=H0.5(σ(ℳLR)),,∈ℝK×r,σ(M_LR)= (UV )+1, _eff=H_0.5\! (σ(M_LR) ), ,V ^K× r, where H0.5H_0.5 denotes the same threshold gate used by CausalBindSP. Unless otherwise stated, the rank scan uses K=128K=128, dc=1024d_c=1024, L=4L=4, seed 1, and λ=10−4λ=10^-4. The mask-parameter count per branch is 2Kr2Kr, compared with K2K^2 for the dense CausalBindSP mask. Table A15: CausalBindLR rank scan, seed=1, all three checkpoint protocols. All rows use K=128K=128, dc=1024d_c=1024, L=4L=4, and λ=10−4λ=10^-4. r Params/branch Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 1 256 last 0.935 0.720 45.86 0.630 0.087 8.41 best_auc 0.946 0.756 48.78 0.636 0.079 6.53 best_bedroc 0.935 0.724 46.28 0.632 0.089 8.56 4 1,024 last 0.934 0.731 47.14 0.632 0.085 7.45 best_auc 0.933 0.710 45.14 0.633 0.080 7.67 best_bedroc 0.938 0.744 47.95 0.632 0.078 6.52 8 2,048 last 0.941 0.756 48.67 0.626 0.085 7.60 best_auc 0.941 0.756 48.38 0.634 0.079 6.61 best_bedroc 0.940 0.750 48.29 0.627 0.083 7.32 16 4,096 last 0.933 0.725 46.39 0.630 0.075 6.27 best_auc 0.928 0.693 44.07 0.616 0.075 6.01 best_bedroc 0.932 0.717 45.82 0.630 0.067 5.55 32 8,192 last 0.939 0.738 47.13 0.622 0.085 7.25 best_auc 0.941 0.741 47.44 0.624 0.083 6.75 best_bedroc 0.933 0.705 45.11 0.611 0.075 5.88 64 16,384 last 0.937 0.748 48.37 0.625 0.081 6.83 best_auc 0.936 0.731 46.88 0.622 0.074 5.58 best_bedroc 0.936 0.743 47.79 0.625 0.075 6.58 Table A15 shows that very small ranks are already competitive. Rank r=1r=1 is the most parameter-efficient setting and reaches the best LIT-PCBA EF@1% under best_bedroc, while r=8r=8 gives the strongest DUD-E early-enrichment numbers. Increasing the nominal rank beyond this range does not provide a consistent gain, suggesting that the useful interaction signal is much lower-dimensional than the full K×K×K mask. A6.5.2 Multi-Seed Comparison and Ablation We next compare the most relevant CausalBindLR settings across three seeds under the fixed-endpoint last protocol. The default row uses r=1r=1 with λspaLR=0 _spa^LR=0; the +Lspa+L_spa row adds the optional mask penalty. The table also includes the main CausalBindSP multi-seed reference so that the low-rank implementation can be read against the main model directly. Table A16: Multi-seed last-checkpoint comparison for CausalBindLR and the main CausalBindSP reference. Method n DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% CausalBindLR (r=1r=1) 3 0.937±0.0040.937±0.004 0.732±0.0110.732±0.011 46.87±0.9146.87±0.91 0.630±0.0060.630±0.006 0.085±0.0060.085±0.006 7.22±0.317.22±0.31 CausalBindLR (r=1r=1) +Lspa+L_spa 3 0.935±0.0030.935±0.003 0.727±0.0070.727±0.007 46.39±0.5846.39±0.58 0.623±0.0070.623±0.007 0.083±0.0050.083±0.005 7.51±0.947.51±0.94 CausalBindLR (r=8r=8) +Lspa+L_spa 3 0.940±0.0010.940±0.001 0.750±0.0110.750±0.011 48.30±0.9748.30±0.97 0.621±0.0050.621±0.005 0.087±0.0030.087±0.003 7.73±0.407.73±0.40 CausalBindSP main 4 0.938±0.0040.938±0.004 0.746±0.0040.746±0.004 47.99±0.3947.99±0.39 0.631±0.0060.631±0.006 0.088±0.0070.088±0.007 7.55±0.737.55±0.73 The key observation is that CausalBindLR with r=1r=1 essentially matches the main dense-mask CausalBindSP within one standard deviation while using only 256256 mask parameters per branch instead of 16,38416,384. Adding LspaL_spa to the rank-one model changes little, indicating that the hard rank constraint already provides the structural compression that the L1L_1 term induces only indirectly in the dense-mask version. Rank r=8r=8 improves DUD-E, but the LIT-PCBA early-enrichment metrics are already close to saturated at rank one. A6.5.3 Hyperparameter Robustness The rank-one implementation also changes how the model behaves as the concept count grows. In the dense CausalBindSP mask, increasing K expands the mask as K2K^2 and can over-parameterize the interaction layer. In CausalBindLR with r=1r=1, the mask has only 2K2K parameters per branch, so increasing K mainly changes the number of available concepts rather than the rank of their interaction. Table A17: CausalBindLR K-scan with r=1r=1, seed=1, all three protocols. All rows use dc=1024d_c=1024, L=4L=4, and λ=10−4λ=10^-4. K Params/branch Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 64 128 last 0.935 0.704 44.87 0.632 0.103 8.84 best_auc 0.934 0.702 44.63 0.638 0.099 9.03 best_bedroc 0.936 0.716 45.92 0.631 0.095 6.54 128 256 last 0.935 0.720 45.86 0.630 0.087 8.41 best_auc 0.946 0.756 48.78 0.636 0.079 6.53 best_bedroc 0.935 0.724 46.28 0.632 0.089 8.56 256 512 last 0.935 0.734 47.16 0.622 0.074 5.82 best_auc 0.936 0.744 47.91 0.617 0.077 6.79 best_bedroc 0.938 0.740 47.77 0.629 0.066 5.72 Table A17 shows that the rank-one constraint protects the model from the large-K collapse observed in the dense concept-pair sweep: K=256K=256 remains strong on DUD-E instead of failing under an oversized K2K^2 mask. At the other end, K=64,r=1K=64,r=1 gives the best LIT-PCBA early-enrichment numbers in this scan, suggesting that the property-debiased benchmark can be served by a very small interaction subspace. A6.5.4 Mask Analysis Finally, we analyze whether the explicit rank capacity is actually used by the optimizer. The diagnostics use the same thresholded effective mask and singular-value energy criteria as App. A6.1.7. Table A18: Mask diagnostics for the CausalBindLR rank scan. The reported branch uses the same protocol as Table A9. r capacity σ(ℳ)¯ σ(M) %zeros r@95%r@95\% r@99%r@99\% Interpretation 1 0.86 0.0% 1 1 rank-one by construction 4 0.63 0.0% 1 1 trained solution collapses to rank one 8 0.54 8.9% 1 1 some thresholded zeros, still rank one 16 0.51 23.3% 1 1 more zeros, same effective rank 32 0.51 34.3% 1 1 unused rank capacity 64 0.50 39.7% 1 1 unused rank capacity CausalBindSP dense ref. 0.76 0.0% 1 9–14 rank-one at 95%, richer 99% tail Across all nominal capacities r∈1,4,8,16,32,64r∈\1,4,8,16,32,64\, the trained CausalBindLR mask has r@95%=r@99%=1r@95\%=r@99\%=1. This provides a useful check on the theory–practice story: the low-rank behavior observed in the dense CausalBindSP mask is not merely an artifact of the tanh+L1 +L_1 parameterization, because an explicitly rank-constrained model also converges to a rank-one interaction. A6.6 CausalBindEMB: Experimental Results and Diagnostics The implementation and parameter settings of CausalBindEMB are given in App. A5.4, and the three-way architectural comparison is summarized in App. A5.5. Here we only report its retrieval behavior and mask diagnostics. This keeps A5 focused on implementation and keeps the present A6 section focused on evidence. A6.6.1 Hyperparameter Study on K and dcd_c We first report the embedding-level K/dcK/d_c sweep that determines the representative CausalBindEMB configuration used below. Unlike CausalBindSP and CausalBindLR, this implementation applies the mask after concept pooling, so these scans should be read as pooled-space sensitivity analyses rather than concept-pair support analyses. Table A19: CausalBindEMB K-scan (fix dc=256d_c=256), seed=1, all three protocols. K Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 32 last 0.942 0.743 47.35 0.638 0.090 7.48 best_auc 0.942 0.747 47.84 0.640 0.085 6.86 best_bedroc 0.937 0.716 45.79 0.643 0.066 4.38 64 last 0.939 0.754 48.56 0.636 0.102 9.60 best_auc 0.941 0.765 49.47 0.635 0.097 8.18 best_bedroc 0.939 0.751 48.33 0.636 0.101 9.25 128 last 0.935 0.715 45.62 0.623 0.082 6.50 best_auc 0.938 0.713 45.29 0.630 0.084 7.50 best_bedroc 0.938 0.701 44.49 0.630 0.080 7.01 256 last 0.936 0.737 47.55 0.624 0.080 7.08 best_auc 0.937 0.739 47.69 0.623 0.080 7.39 best_bedroc 0.934 0.728 46.92 0.624 0.081 7.04 Table A20: CausalBindEMB dcd_c-scan (fix K=64K=64), seed=1, all three protocols. dc=256d_c=256 is strongest on LIT-PCBA early enrichment. dcd_c Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% 128 last 0.943 0.732 47.09 0.606 0.074 5.88 best_auc 0.944 0.726 46.59 0.617 0.075 6.97 best_bedroc 0.943 0.738 47.49 0.602 0.071 6.03 256 last 0.939 0.754 48.56 0.636 0.102 9.60 best_auc 0.941 0.765 49.47 0.635 0.097 8.18 best_bedroc 0.939 0.751 48.33 0.636 0.101 9.25 512 last 0.942 0.750 48.32 0.624 0.079 6.45 best_auc 0.945 0.747 48.11 0.625 0.078 6.68 best_bedroc 0.943 0.753 48.53 0.619 0.079 6.43 1024 last 0.940 0.745 48.01 0.622 0.088 8.12 best_auc 0.943 0.760 48.82 0.628 0.090 8.47 best_bedroc 0.935 0.716 45.73 0.622 0.077 6.92 The embedding-level sweeps are smoother than the concept-pair sweeps, but the best operating point is still concentrated around moderate concept count and intermediate channel width. We therefore use K=64K=64, dc=256d_c=256, and λ=10−2λ=10^-2 as the representative CausalBindEMB configuration in Table A21. A6.6.2 Experimental Results Table A21: Three-protocol comparison among HypSeek, CausalBindSP, and CausalBindEMB. CausalBindSP uses K=128K=128, dc=1024d_c=1024, λ=10−4λ=10^-4, L=4L=4, which is the main operating point analyzed in the paper. CausalBindEMB uses K=64K=64, dc=256d_c=256, λ=10−2λ=10^-2, selected as the strongest setting in the embedding-level sweep. All rows are seed-1 results. Model Protocol DUD-E LIT-PCBA AUROC BEDROC80.5 EF@1% AUROC BEDROC80.5 EF@1% HypSeek [Wang et al., 2026] last 0.909 0.605 37.83 0.603 0.061 4.62 best_auc 0.913 0.629 39.63 0.608 0.063 5.13 best_bedroc 0.923 0.676 42.71 0.607 0.063 4.56 CausalBindSP last 0.935 0.744 47.69 0.639 0.097 8.56 best_auc 0.934 0.722 46.41 0.628 0.074 5.84 best_bedroc 0.935 0.747 48.24 0.643 0.094 8.42 CausalBindEMB last 0.939 0.754 48.56 0.636 0.102 9.60 best_auc 0.941 0.765 49.47 0.635 0.097 8.18 best_bedroc 0.939 0.751 48.33 0.636 0.101 9.25 Main comparison. Table A21 now puts the three relevant models under the same three checkpoint-selection protocols, so the comparison is directly aligned with the earlier protocol analyses in this appendix. The two CausalBind variant rows use the representative configurations selected earlier in their respective sweeps: CausalBindSP is the paper’s main operating point, while CausalBindEMB is the strongest embedding-level setting identified by the K/dcK/d_c scan. Two patterns stand out. First, both CausalBind variants remain clearly stronger than HypSeek across all three protocols, especially on LIT-PCBA early-enrichment metrics. This confirms that the gain is not tied to one specific checkpoint rule. Second, the sparse concept-pair and embedding-level variants emphasize slightly different strengths: CausalBindSP gives the best LIT-PCBA AUROC under last/best_bedroc, whereas CausalBindEMB is especially strong on LIT-PCBA BEDROC80.5 / EF@1% under last, and also gives the best DUD-E numbers under best_auc. What this means empirically. The purpose of this section is not to argue that CausalBindEMB should replace the main model, but to show that the sparse/low-rank interaction prior is effective beyond a single implementation. CausalBindSP remains the paper’s primary architecture because it offers a cleaner causal interpretation and aligns more directly with the concept-pair theory. CausalBindEMB, however, demonstrates that a simpler pooled-space realization can also produce strong retrieval, which makes the overall design principle more credible. A6.6.3 Theory Correspondence The theory correspondence of the embedding-level variant is simpler, but also narrower, than for the main concept-level model. Because the mask acts on pooled embeddings rather than on explicit concept pairs, it does not naturally match the sparse-support picture behind Condition 2.i and Theorem 2. In particular, there is no direct analogue of concept-pair support geometry or the antichain-style condition discussed for the concept-level mask. For that reason, this implementation should not be presented as evidence for the sparse identifiability theorem. By contrast, the connection to the low-rank theorem is much more natural. The embedding-level mask is exactly a constrained interaction operator in a dcd_c-dimensional shared space, so asking whether useful solutions are low-rank is directly meaningful here. In that sense, the variant is much closer in spirit to Condition 3.i and Theorem 3: it tests whether a small number of interaction modes in the pooled cross-modal space is already sufficient for strong retrieval. Table A22: Spectral diagnostics for the embedding-level mask (K=64K=64, dc=256d_c=256, λ=10−2λ=10^-2, seed 1). We report the two branchwise product masks used in the paper’s interaction view, evaluated under the last checkpoint. Branch %zeros r@95%r@95\% r@99%r@99\% mol–pocket 0.0% 1 46 mol–sequence 0.0% 1 46 Table A22 makes the interpretation more concrete. The embedding-level mask exhibits no exact zeros after gating, so it provides no support for the sparse-support assumptions behind Theorem 2. At the same time, its spectral profile is clearly compressed: for both branches, a single dominant mode already explains 95%95\% of the squared singular-value energy, and only 4646 modes are needed to explain 99%99\% out of the full ambient dimension dc=256d_c=256. This is not a strict low-rank matrix in the algebraic sense, but it is a strongly effectively low-rank interaction operator. We also checked nearby late checkpoints (epochs 40, 45, and 50), and the same r@95%=1r@95\%=1, r@99%=46r@99\%=46 pattern persists, so this is not a single-checkpoint artifact. Across all scanned configurations, the embedding-level variant is also somewhat smoother than the concept-level one: its DUD-E AUROC varies within a relatively narrow band (0.9350.935–0.9430.943, n=7n=7), whereas the concept-level sweeps in Tables A5 and A6 span a wider range. This difference is plausible architecturally. The embedding-level mask acts after pooling and therefore modulates a coarser representation; the concept-level mask, in contrast, directly shapes the fine-grained interaction structure before pooling. Taken together, these observations suggest a clean division of roles: the concept-level model is the better vehicle for the paper’s causal and sparse-support story, while the embedding-level alternative provides additional empirical support for the low-rank interaction view. A6.7 Overall Observations Across Three Implementations What remains stable across the sweep. The most important observation is that the main qualitative conclusion does not depend on a single hyperparameter choice, checkpoint protocol, or implementation variant. Across the main CausalBindSP comparison (Table A4), the explicit low-rank results (Tables A15–A18), and the embedding-level alternative (Tables A19–A21), constrained cross-modal interaction consistently improves over dense holistic alignment on the retrieval metrics emphasized in the main text, especially early enrichment on LIT-PCBA. How the three implementations differ. CausalBindSP is the main paper model: it keeps explicit concept-pair structure and is therefore the cleanest implementation for studying sparse support, mask calibration, and concept-level interpretation. CausalBindLR is the direct low-rank concept-basis implementation: it matches the main dense-mask model with far fewer mask parameters and shows that rank-one or near-rank-one interaction is often sufficient. CausalBindEMB is a pooled-space alternative: it can be very strong empirically, especially on LIT-PCBA early enrichment, but its mask acts after concept pooling and therefore has a weaker connection to component-wise concept identifiability. What the hyperparameter studies collectively suggest. The CausalBindSP K/dc/λ/LK/d_c/λ/L scans point to a coherent operating regime: moderate concept granularity, adequate channel width, and moderate interaction depth work best, while overly large dense concept-pair masks can become brittle. The CausalBindLR scans sharpen this lesson by showing that explicit rank control protects against the large-K collapse and that increasing rank beyond a small value rarely helps. The CausalBindEMB scans are smoother, but the strongest setting is still a moderate pooled representation (K=64K=64, dc=256d_c=256). How the sweep connects back to theory. Taken together, the three implementations separate form from empirical regime. CausalBindSP has the sparse concept-pair form of Theorem 2, but its strongest λ=10−4λ=10^-4 setting behaves more like a low-rank solution; at λ=10−3λ=10^-3 it moves closer to sparse support. CausalBindLR directly realizes the subspace-identifiability form of Theorem 3 and confirms that the learned interaction can be rank one in practice. CausalBindEMB supports the broader low-rank interaction principle but should be interpreted as a pooled-space variant rather than a component-wise concept-identifiability result. A6.8 Discussion on Theory–Practice Gap The theoretical results in Section 2 are intentionally sharp: Theorem 2 describes what becomes identifiable under the sparse-support assumptions, while Theorem 3 gives a subspace-identifiability statement under low-rank interaction. The experiments in App. A6 show that these two theorems are both useful for interpretation, but not in the same way. This subsection makes that distinction explicit by connecting the compact mask statistics in Table 3 with the branchwise diagnostics in Table A9. Main takeaway. The central lesson is not that the learned interaction becomes exactly element-wise sparse. Rather, the causal prior pushes training toward structurally simpler cross-modal interactions: stronger sparsification moves CausalBindSP toward the sparse-support picture, while milder sparsification and explicit CausalBindLR reveal a low-rank regime. Thus low-rank subspace identifiability is not merely a secondary theoretical case, but an important description of the regime that modern neural training naturally reaches. What the experiments do support. The experiments support both theorem regimes, but through different implementations and operating points. In Table 3, ‘L1L_1 avg’ is the average absolute mask entry, %zeros is the fraction of entries thresholded to exact zero, and r@95%r@95\%/r@99%r@99\% are effective ranks explaining 95%95\%/99%99\% of squared singular-value energy. For CausalBindSP, λ=10−3λ=10^-3 yields strong DUD-E retrieval together with meaningful exact sparsity (20.5%20.5\% zeros in the compact table), matching the sparse-support picture of Theorem 2; the default λ=10−4λ=10^-4 setting remains dense entry-wise (0%0\% zeros) but is strongly low-rank in the dominant energy component (r@95%=1r@95\%=1), matching Theorem 3. For λ≥10−2λ≥ 10^-2, the mask enters an excessive hard-zero regime and performance degrades sharply, so sparsity alone is not the desired behavior. In CausalBindLR, the rank constraint is imposed directly and the learned masks still collapse to effective rank one, which independently supports the low-rank/subspace interpretation. In CausalBindEMB, the same low-rank tendency appears after pooling, but this evidence is architectural rather than component-wise. Why this is not a contradiction. The theorem statements are intentionally sharp, whereas optimization can realize more than one useful structural regime. A dense K×K×K mask with an L1L_1 penalty can behave as sparse support under stronger regularization or as a low-rank operator under milder regularization; an explicit rank-factorized mask enforces the latter by construction. The empirical message is therefore not that one theorem replaces the other, but that the two theorem regimes identify different useful limits of the same causal interaction prior. How we position the current paper. For this reason, we position CausalBind as a causally motivated structural prior whose empirical behavior is consistent with both theoretical identifiability results. The theory explains why dense holistic interaction is the wrong object, why structural constraints are needed at all, and why sparse-support and low-rank regimes are the right two reference points. The experiments then show how these reference points appear in practice: CausalBindSP connects most directly to sparse concept-pair structure, CausalBindLR directly validates the subspace-identifiability regime, and CausalBindEMB shows that a weaker pooled-space version of the same principle can also work well. Appendix A7 Limitations, Broader Impact, and Usage of LLMs. Limitation on scope of the current evidence. This paper should be read primarily as a study of retrieval-based virtual screening under cross-dataset generalization, not as a complete evaluation of protein-molecule modeling in every downstream regime. Accordingly, our main evidence primarily comes from two retrieval benchmarks, DUD-E and LIT-PCBA, while App. A6.1.9 provides a supplementary Merck-FEP-8 affinity-ranking transfer check; the strongest claims still concern early enrichment rather than absolute affinity calibration. This scope is deliberate: the central question of the paper is whether replacing dense holistic alignment with sparse or low-rank cross-modal interaction improves the high-throughput screening stage. It also means that broader claims should be made cautiously. The current experiments do not yet establish how uniformly the gain transfers across protein families, scaffold splits, time-stratified settings, or much larger benchmark collections. Broader impact and misuse clarification. Better virtual screening can reduce the cost of early discovery by improving which candidates are advanced to docking and experimental follow-up, potentially accelerating the identification of useful therapeutics. At the same time, any method that improves prioritization of protein-molecule interactions could in principle be misused to search for ligands against harmful targets. Our model is less directly dual-use than a generative molecular-design system, because it is a scoring and ranking tool rather than a molecule generator, but this does not remove the need for safeguards. In practical use, its predictions should be treated as one stage in a controlled pipeline and paired with downstream safety, ADMET, toxicity, and experimental validation filters before any wet-lab deployment. Usage of LLMs. LLMs were used to assist with writing and editing, but they are not the source of any scientific claims or empirical results in this paper. All technical content is derived from our own theoretical analysis and experimental work, and all key numbers are directly traceable to our codebase and training logs. We did not use LLMs to generate scientific hypotheses. Their role was limited to language polishing and formatting assistance, which we carefully reviewed and edited for accuracy and clarity. Appendix A8 Reproducibility and Asset Provenance Code basis and development path. The implementation of CausalBind was developed by extending the retrieval pipeline used in HypSeek and cross-checking evaluation behavior against LigUnity. Concretely, our code keeps the same broad three-stream retrieval setting as HypSeek, while replacing the dense shared-space interaction with the concept extractor and sparse/low-rank cross-modal masking modules described in Section 3. In this sense, the present implementation should be read as a method-development layer on top of existing retrieval infrastructure rather than as a from-scratch rewrite of every upstream component. Main software assets. The most important inherited software assets are the HypSeek code base, the LigUnity code base used for comparison and protocol alignment, the Uni-Mol/Uni-Core stack used by the molecular and pocket branches, and the ESM-2 protein language model used by the sequence branch. In our local development workflow, many of the inherited Uni-Mol-style source files explicitly carry MIT license headers, and the HypSeek repository itself is marked as MIT-licensed in its accompanying README. The LigUnity repository explicitly distinguishes code and data licenses in its README, with code released under Apache 2.0 and the accompanying data resources released under C BY-NC 4.0. These upstream assets are credited throughout the paper via citations to HypSeek, LigUnity, Uni-Mol, Uni-Core, and ESM-2. Data assets. The training and evaluation data are not newly created by this paper. As described in Section 4 and Appendix A5, training follows the same assay-level setup used by LigUnity and HypSeek, drawing from ChEMBL, BindingDB, and PDBBind, where ChEMBL/BindingDB assays whose target UniProt IDs appear in the DUD-E, LIT-PCBA, or DEKOIS 2.0 benchmarks are removed. The main evaluation benchmarks are DUD-E and LIT-PCBA, and the appendix additionally reports transfer results on Merck-FEP-8. These are all existing public or previously published resources, and they remain subject to the terms, access conditions, and redistribution rules set by their original providers. Our submission does not re-license any of these datasets. What is provided for reproducibility. The code is publicly available at https://github.com/lokali/CausalBind. The repository contains the CausalBind implementation (the core model, loss, and task files for all variants), scripts that download the public training data, benchmarks, and pretrained backbones from their original sources, training and evaluation entry points with the exact configurations reported in the paper, and the tools for the mask-structure analysis, the DEKOIS 2.0 out-of-distribution evaluation, and the interpretability case study. We also release the trained checkpoints of the reported models together with a one-command script that reproduces the DUD-E and LIT-PCBA results. Processed datasets are not redistributed; they remain subject to the terms of their original providers. Scope of the current license discussion. Our goal in this appendix section is to make the provenance of the main assets explicit and to clarify which parts of the project are inherited versus newly added by CausalBind. We do not claim to re-publish all upstream assets under a unified new license. Instead, the intended reading is modular: the new CausalBind code introduced in this paper is layered on top of existing open-source and public research assets, and each inherited component should continue to be used under its original license or terms of use.