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DiMOS: Doob-Guided Inference-Time Multi-Objective Search for Scientific Design
Ziqing Wang, Qijie Zhu, Weimin Wu, Zeqi Ye, Minshuo Chen, Han Liu, Kaize Ding
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Abstract
Abstract:Scientific design often requires jointly satisfying multiple objectives and constraints. Pretrained masked diffusion models provide a generative foundation for this task, but fine-tuning them to meet these objectives and constraints incurs additional training costs, motivating inference-time guidance with frozen models. However, such guidance faces two challenges: pass-or-fail constraints and black-box reward models may provide no useful gradients, while jointly satisfying multiple requirements can leave a small feasible region, making feasible designs difficult to find within a limited inference budget. To address these challenges, we introduce DiMOS, a training-free framework for multi-objective scientific design. Using joint rewards from candidate completions, DiMOS performs approximate Doob-guided local resampling without requiring reward gradients. To allocate computation efficiently, it uses budget-efficient trajectory search to focus computation on promising continuations. Across six DNA, protein, and RNA tasks, DiMOS attains the highest joint success rate at comparable generation times, up to $1.98\times$ the strongest baseline on DNA and protein, while maintaining high sequence uniqueness and naturalness.
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- Source: https://arxiv.org/abs/2610.05808v1
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DiMOS: Doob-Guided Inference-Time Multi-Objective Search for Scientific DesignThanks: Corresponding authors. Ziqing Wang †thanks: Equal contribution. Qijie Zhu11footnotemark: 1 Weimin Wu11footnotemark: 1 Zeqi Ye Affiliation: Minshuo Chen Han Liu Kaize Ding22footnotemark: 2 Affiliation: Northwestern University Abstract Scientific design often requires jointly satisfying multiple objectives and constraints. Pretrained masked diffusion models provide a generative foundation for this task, but fine-tuning them to meet these objectives and constraints incurs additional training costs, motivating inference-time guidance with frozen models. However, such guidance faces two challenges: pass-or-fail constraints and black-box reward models may provide no useful gradients, while jointly satisfying multiple requirements can leave a small feasible region, making feasible designs difficult to find within a limited inference budget. To address these challenges, we introduce DiMOS, a training-free framework for multi-objective scientific design. Using joint rewards from candidate completions, DiMOS performs approximate Doob-guided local resampling without requiring reward gradients. To allocate computation efficiently, it uses budget-efficient trajectory search to focus computation on promising continuations. Across six DNA, protein, and RNA tasks, DiMOS attains the highest joint success rate at comparable generation times, up to 1.98×1.98× the strongest baseline on DNA and protein, while maintaining high sequence uniqueness and naturalness. Figure 1: Joint success and uniqueness. Bubble area encodes within-task normalized 3-mer correlation (DNA), scRMSD <2Å<2\, rate (protein), and Optimus-MRL (RNA). Three-seed means. 1 Introduction Scientific sequence design is a central task in synthetic biology and therapeutic development (Zrimec et al., 2022; Cao et al., 2022). In practice, however, scientists often need to satisfy several requirements simultaneously rather than improve a single property in isolation (Stanton et al., 2022; Gruver et al., 2023). For example, regulatory DNA should combine target-cell activity with limited off-target activity (Gosai et al., 2024), while therapeutic mRNA requires both efficient translation and sufficient stability (Leppek et al., 2022). We therefore focus on multi-objective scientific sequence design, seeking candidate sequences that jointly meet multiple objectives and constraints. Masked diffusion models provide a generative foundation for this task by learning sequence distributions through the reconstruction of masked tokens (Sahoo et al., 2024; Wang et al., 2025). However, sampling from a learned distribution does not ensure that designs meet the specified objectives (Li et al., 2025). Fine-tuning-based methods address this gap by updating pretrained models toward task-specific rewards (Black et al., 2024; Clark et al., 2024; Wang et al., 2025; Han et al., 2026; Domingo-Enrich et al., 2025), but the added training becomes a burden as objectives or thresholds change (Li et al., 2025). This motivates inference-time guidance that reuses a frozen generative prior while steering generation toward different combinations of objectives and constraints. Such guidance needs one signal to follow, so we assign each candidate a joint reward R(x)R(x) that is high only when every requirement is met. Yet guiding a frozen model with such a reward presents two challenges: • ❶How can we guide generation when R is non-differentiable? Scientific design often requires improving one property while enforcing hard constraints on others. Even when reward models are differentiable, pass-or-fail constraints provide no useful gradient signal for correcting constraint violations. The reward model itself can also be non-differentiable: black-box tools such as ViennaRNA assess RNA folding without directly providing sequence gradients (Lorenz et al., 2011). Gradient-based inference-time methods therefore struggle to follow such a reward (Wu et al., 2023; Nisonoff et al., 2025). • ❷How can we effectively search for jointly feasible designs within a given inference budget? Gradient-free methods handle non-differentiable feedback but often require extensive sampling and repeated reward evaluations (Li et al., 2025; Singhal et al., 2025). Meanwhile, simultaneously satisfying multiple objectives and constraints can leave only a small feasible region, particularly when objectives conflict, further increasing the search effort required (Stanton et al., 2022). Efficiently allocating computation to discover jointly feasible candidates within a limited inference budget is therefore another key challenge. To address these challenges, we propose DiMOS, a training-free framework for multi-objective scientific design with frozen masked diffusion models (Fig. 2). Its approximate Doob-guided local resampling steers generation using reward evaluations alone, accommodating non-differentiable feedback. Building on this local guidance, budget-efficient trajectory search directs computation toward promising continuations to find jointly feasible designs. Our three contributions are: • Framework. We propose DiMOS, a Doob-guided framework for inference-time multi-objective design with frozen masked diffusion models. It supports black-box objectives and hard constraints through evaluations of R, keeping the generative model frozen and requiring reward values alone. • Algorithm. We develop a budget-efficient trajectory search algorithm that combines stage-wise branching with reward-guided refinement. It allocates computation across local candidate comparisons and promising continuations to improve joint success within a given inference budget. • Effectiveness. Across six multi-objective tasks spanning DNA, protein, and RNA design (Fig. 1), DiMOS attains the highest joint success rate under matched wall-clock budgets, up to 1.98×1.98× the strongest baseline on DNA and protein, while maintaining sequence naturalness. 2 Related Work Multi-Objective Scientific Design. Multi-objective scientific design seeks sequences with several desired properties. Discrete diffusion supports regulatory DNA design (Sarkar et al., 2024). Constrained reinforcement learning adapts generative models to optimize target-cell activity while controlling off-target effects (Chen et al., 2025b). Other approaches explore Pareto trade-offs through multi-objective Bayesian optimization (Stanton et al., 2022), peptide tree search (Tang et al., 2025), or rank-directional flow guidance (Chen et al., 2025a). Pareto optimality alone, however, does not ensure simultaneous satisfaction of application thresholds. Our focus is finding designs that jointly meet specified property thresholds through inference-time search. Training-Free Adaptation. Training-free methods steer a frozen generative prior at sampling time using reward feedback. (1) Gradient guidance modifies reverse transitions using reward gradients or gradient-based approximations (Nisonoff et al., 2025; Schiff et al., 2025). (2) Particle methods reweight and resample trajectories, sometimes using gradient-guided proposals (Wu et al., 2023; Singhal et al., 2025). (3) Evaluation-based guidance ranks or resamples candidate continuations using reward values or estimated future rewards (Guo et al., 2024; Li et al., 2025), with iterative refinement wrapping single-shot guidance in an outer noising-and-denoising loop (Uehara et al., 2025a). That outer loop is orthogonal to the per-step mechanisms studied here and could wrap any of them. Related probabilistic formulations connect adaptation to Doob’s h-transform, soft value functions, and posterior sampling (Uehara et al., 2025b). These include learned corrections or samplers (Denker et al., 2024; Rector-Brooks et al., 2025), as well as training-free inference for image editing (Nguyen et al., 2025) and adaptation with generic rewards (Zhu et al., 2026). DiMOS uses reward evaluations alone for approximate Doob-guided local resampling, then concentrates computation on promising continuations via shared prefixes and branch selection. 3 Preliminaries We introduce discrete diffusion (Austin et al., 2021) in Sec. 3.1 and Doob’s h-transform in Sec. 3.2. 3.1 Discrete Diffusion Let =NX=V^N be the space of length-N sequences over a finite alphabet V. A forward continuous-time Markov chain (Yt)t∈[0,T](Y_t)_t∈[0,T] corrupts Y0∼pdataY_0 p_data on a finite state space ⊇S . We use the column convention: for y≠xy≠ x, Qt(y,x)≥0Q_t(y,x)≥ 0 is the jump rate from x to y. The marginal probability vector ptp_t evolves as dptdt=Qtpt,Qt(x,x)=−∑y≠xQt(y,x),p0=pdata. dp_tdt=Q_tp_t, Q_t(x,x)=- _y≠ xQ_t(y,x), p_0=p_data. (1) The corruption process is chosen so that pTp_T approaches a tractable reference distribution. Writing denoising time as u=T−tu=T-t, define the reverse-time process Y~u:=YT−u Y_u:=Y_T-u, so that denoising runs from u=0u=0 to u=Tu=T, corresponding to the forward-time direction t=Tt=T to t=0t=0. Its exact off-diagonal generator is Q~u(y,x)=QT−u(x,y)pT−u(y)pT−u(x),y≠x, Q_u(y,x)=Q_T-u(x,y) p_T-u(y)p_T-u(x), y≠ x, (2) for states with pT−u(x)>0p_T-u(x)>0. Thus, starting from Y~0∼pT Y_0 p_T, the reverse process progressively removes the corruption introduced by the forward process and recovers Y~T∼p0=pdata Y_T p_0=p_data. The reverse dynamics in Eq. (2) depend on the density ratios pt(y)/pt(x)p_t(y)/p_t(x), which characterize how the reverse process should move between discrete states (Lou et al., 2024). Masked Discrete Diffusion. A commonly used specialization of discrete diffusion for sequence generation is a masked discrete diffusion model (Sahoo et al., 2024; Shi et al., 2024). For sequence generation, we augment the token alphabet V with a mask symbol ∉ m and let ~=∪ V=V∪\ m\ and =~NS= V^N. Masked discrete diffusion independently replaces each clean token by m at a time-dependent rate λt _t (Sahoo et al., 2024; Shi et al., 2024). Writing αt=exp(−∫0tλrdr) _t= (- _0^t _r\,dr) for the probability that a coordinate remains unmasked by time t, the forward transition from x0∈x_0 is q0,t(x∣x0)=∏j=1N[αtxj=x0j+(1−αt)xj=].q_0,t(x x_0)= _j=1^N [ _t1\x^j=x_0^j\+(1- _t)1\x^j= m\ ]. (3) Thus, the forward process progressively replaces clean tokens by masks and never changes one clean token directly into another. As αT→0 _T→ 0, pTp_T approaches this fully masked reference distribution δN _ m^N. For a masked coordinate xj=x^j= m, let xj←ax^j← a denote the state obtained by replacing that coordinate with a∈a . Under the masking kernel, pt(xj←a)pt(x)=αt1−αtℙ(Y0j=a∣Yt=x). p_t(x^j← a)p_t(x)= _t1- _tP(Y_0^j=a Y_t=x). Thus the reverse dynamics are parameterized by the clean-token posterior. We approximate this posterior with μθ,tj(⋅∣x)∈Δ() _θ,t^j(· x)∈ (V), where Δ() (V) is the simplex over V, yielding Q^θ,T−t(xj←a,x)=λtαt1−αtμθ,tj(a∣x). Q_θ,T-t (x^j← a,x )= _t _t1- _t _θ,t^j(a x). At inference time, we apply a token temperature η>0η>0 to the denoiser distribution and define μθ,tj,η(a∣x)=μθ,tj(a∣x)1/η∑a′∈μθ,tj(a′∣x)1/η. _θ,t^j,η(a x)= _θ,t^j(a x)^1/η _a _θ,t^j(a x)^1/η. (4) Thus, η=1η=1 recovers the denoiser distribution, while η<1η<1 sharpens it and η>1η>1 flattens it. The reverse process is simulated on a finite denoising grid 0=t0<t1<⋯<tL=T0=t_0<t_1<·s<t_L=T. Discretizing the reverse dynamics over each interval [tℓ−1,tℓ][t_ -1,t_ ] gives a transition kernel Pθ,tℓ(y∣x):=ℙθ(Xtℓ−1=y∣Xtℓ=x).P_θ,t_ (y x):=P_θ (X_t_ -1=y X_t_ =x ). For masked diffusion, this transition progressively reveals masked coordinates according to the prescribed noise schedule. Newly revealed tokens are sampled from μθ,tℓj,η(⋅∣x) _θ,t_ ^j,η(· x), while previously revealed tokens remain fixed. We treat the fixed sampling temperature η as part of the finite-step sampling procedure and suppress it in the notation Pθ,tℓP_θ,t_ . We write ℙθP_θ for the resulting path law and pθp_θ for its terminal distribution at t0t_0. 3.2 Diffusion Guidance via Doob’s h-Transform Reward-guided diffusion reweights the terminal distribution of a pretrained model (Wu et al., 2023; Li et al., 2025). Following the reward-tilting perspective in KL-regularized reinforcement learning and stochastic control (Kappen, 2005; Levine, 2018), we parameterize the terminal weight by the reward function R:→ℝR:X of Sec. 1. Definition 1 (Guided Terminal Distribution). Given a reward function R:→ℝR:X and a reward temperature β>0β>0, the corresponding guided terminal distribution is πR(x0)=pθ(x0)exp(R(x0)/β)ZR,ZR=X0∼pθ[exp(R(X0)/β)]. _R(x_0)= p_θ(x_0) (R(x_0)/β )Z_R, Z_R=E_X_0 p_θ [ (R(X_0)/β ) ]. (5) We refer to the problem of modifying the reverse dynamics of the pretrained diffusion model so that their terminal marginal is πR _R as terminal-reweighted diffusion guidance. Both soft objectives and hard constraints can be represented through the choice of R. A soft objective assigns a scalar reward R(x)∈ℝR(x) to each design, with larger values indicating greater preference. For a hard constraint specified by a feasible set ⊆A , we use the indicator-based reward function R(x)=−Px∉,P>0.R(x)=-P1\x \, P>0. (6) In the limit P→∞P→∞, exp(R(x)/β) (R(x)/β ) reduces to x∈1\x \, recovering exact conditioning on the feasible set. Thus, soft objectives and hard constraints share the same reward-tilting formulation. Doob’s h-transform provides an exact solution to the terminal-reweighting problem for Markov processes (Rogers and Williams, 2000; Särkkä and Solin, 2019). More generally, for a positive terminal weight ϕ:→ℝ>0φ:X _>0, define the corresponding terminal-reweighted distribution by πϕ(x0)=pθ(x0)ϕ(x0)X0∼pθ[ϕ(X0)]. _φ(x_0)= p_θ(x_0)φ(x_0)E_X_0 p_θ[φ(X_0)]. (7) Definition 1 corresponds to the choice ϕ(x0)=exp(R(x0)/β)φ(x_0)= (R(x_0)/β). To construct a reverse process with terminal distribution πϕ _φ, we propagate the terminal weight ϕφ backward through the original reverse dynamics using Doob’s h-function. Definition 2 (Doob’s h-Function). For the terminal weight ϕφ, define the backward value function hℓ(x)=ℙθ[ϕ(X0)∣Xtℓ=x],h0(x0)=ϕ(x0).h_ (x)=E_P_θ [φ(X_0) X_t_ =x ], h_0(x_0)=φ(x_0). (8) The h-function propagates the terminal preference backward through the original reverse process; it is the soft value function of the corresponding control problem (Uehara et al., 2025b). Using this function, Doob’s h-transform reweights the reverse transitions so as to realize the desired terminal distribution. Proposition 1 (Doob’s h-Transform). Let pθp_θ denote the terminal distribution at t0t_0 of the original reverse sampler initialized from XtL=xLX_t_L=x_L. Define the transformed reverse transition kernel by Pθ,tℓh(y∣x)=Pθ,tℓ(y∣x)hℓ−1(y)hℓ(x).P^h_θ,t_ (y x)=P_θ,t_ (y x) h_ -1(y)h_ (x). (9) Suppose the reverse sampler is initialized from the same fixed initial state XtL=xLX_t_L=x_L as the original diffusion sampler. Then Pθ,tℓhP^h_θ,t_ is a valid Markov transition kernel, and the terminal marginal of the transformed chain is exactly πϕ _φ. The proof of Proposition 1 is deferred to Appendix A. In masked discrete diffusion, the reverse sampler always starts from the same fully masked state XtL=NX_t_L= m^N. Hence, the fixed-initial-state condition of Proposition 1 is naturally satisfied, and Doob’s h-transform changes the terminal marginal from pθp_θ to πϕ _φ by reweighting the pretrained reverse transitions. Figure 2: Overview of DiMOS. Top: multi-objective design with a frozen prior. Middle: K branches from step s, with reward-guided refinement. Bottom: M token candidates are enumerated, completed, and stochastically resampled using prior and reward weights. 4 Method In this section, we introduce DiMOS for multi-objective scientific design using a frozen masked diffusion model. We first express the property requirements as a joint reward (Sec. 4.1). Using this feedback, approximate Doob-guided local resampling compares candidate updates and samples the next state without reward gradients (Sec. 4.2). We then introduce budget-efficient trajectory search to concentrate computation on promising continuations (Sec. 4.3). 4.1 Joint Reward for Multiple Requirements In scientific design, candidates often need to meet multiple property thresholds. We therefore instantiate the feasible-set formulation (hard constraints) of Sec. 3.2 with per-property thresholds, rather than directly maximizing a weighted sum of per-property rewards (soft objectives), which does not in general guarantee joint feasibility. This setting also challenges gradient-based guidance: pass-or-fail constraints provide no useful gradient signal for correcting violations, and reward models may themselves be non-differentiable. We consider n reward functions Rk:→ℝR_k:X , each with an acceptance threshold ckc_k and a direction ϵk∈1,−1 _k∈\1,-1\, where 11 denotes a lower-bound requirement and −1-1 an upper-bound requirement. Define the signed constraint margin dk(x)=ϵk(Rk(x)−ck)d_k(x)= _k(R_k(x)-c_k). Requirement k is satisfied if and only if dk(x)≥0d_k(x)≥ 0. The joint feasible set is therefore =x:dk(x)≥0for allkA=\x:\ d_k(x)≥ 0\ for all\ k\. A single indicator of A assigns the same penalty to every infeasible design, regardless of which requirements are satisfied. To retain feedback about individual requirements, we encode each using a hard or soft gate and aggregate in log space, R(x)=∑k=1nwkgk(dk(x)),gk(d)=−P 1d<0(hard gate),logσ(d/τk)(soft gate),R(x)= _k=1^nw_k\,g_k (d_k(x) ), g_k(d)= cases-P\,1\d<0\&(hard gate),\\[2.0pt] σ(d/ _k)&(soft gate), cases (10) where σ is the logistic sigmoid, wk>0w_k>0 weights each requirement, and τk>0 _k>0 controls the soft gate’s transition width. A hard gate returns 00 when the requirement is satisfied and −P-P otherwise. A soft gate remains negative and increases continuously toward 00 as the margin grows, so it still distinguishes candidates above the threshold. Both gate types use the same feasibility criteria while providing different reward feedback. We retain the finite penalty P>0P>0 of Eq. (6), so violating a hard requirement does not force its weight to zero. Under the exponential tilt, log-space aggregation yields multiplicative per-requirement factors (Appendix D). 4.2 Approximate Doob-Guided Local Resampling Doob’s h-transform connects terminal reward to decisions during denoising. Using ϕ(x0)=exp(R(x0)/β)φ(x_0)= (R(x_0)/β) with the joint reward in Eq. (10), Proposition 1 gives the relative weight of a candidate next state y from the current state x at time tℓt_ , Pθ,tℓh(y∣x)∝Pθ,tℓ(y∣x)hℓ−1(y),P^h_θ,t_ (y x) P_θ,t_ (y x)\,h_ -1(y), (11) since the normalizer hℓ(x)h_ (x) is shared by all candidates. Guidance thus requires relative future values, but evaluating hℓ−1(y)h_ -1(y) entails averaging over all remaining continuations. Completion-Based Value Estimation. To approximate hℓ−1(y)h_ -1(y), we complete the masked state y to a clean sequence x^0(y) x_0(y) using the frozen denoiser, and use its terminal weight as a proxy (Li et al., 2025; Uehara et al., 2025a): h^ℓ−1(y)=ϕ(x^0(y))=exp(R(x^0(y))/β). h_ -1(y)=φ ( x_0(y) )= (R( x_0(y))/β ). (12) Enumeration and Stochastic Resampling. To avoid repeated token proposals, we enumerate distinct alternatives at one newly revealed position. At a guided step tℓt_ , fix x=xtℓx=x_t_ and draw Z∼Pθ,tℓ(⋅∣x)Z P_θ,t_ (· x). Let E=u:xu=,Zu∈E=\u:x^u= m,\,Z^u \ be the reveal set. When E≠∅E≠ , select the position J∈EJ∈ E with the largest denoiser entropy, breaking ties by index, and enumerate the M≤||M≤|V| most probable tokens a1,…,aMa_1,…,a_M under μθ,tℓJ,η(⋅∣x) _θ,t_ ^J,η(· x). For each candidate y(m)=ZJ←amy^(m)=Z^J← a_m, assign logw^m=logμθ,tℓJ,η(am∣x)+logh^ℓ−1(y(m)). w_m= _θ,t_ ^J,η(a_m x)+ h_ -1(y^(m)). (13) We sample m⋆∼Categorical(softmax(logw^1:M))m (softmax( w_1:M)) and retain y(m⋆)y^(m ). Under the factorized token update, all other transition factors are shared; the token prior appears explicitly because alternatives are enumerated rather than sampled from it. Each update uses at most M reward evaluations without reward gradients. We next bound the discrepancy from the full Doob transition, separating context-reweighting, top-M truncation, and value-estimation errors. For this analysis, assume E≠∅E≠ almost surely. The shared context C=(E,(Zu)u∈E∖J)C=(E,(Z^u)_u∈ E \J\) records the reveal set and all newly revealed tokens except the one at J. Its distribution under the base transition is νℓ(c∣x)=ℙZ∼Pθ,tℓ(⋅∣x)(C=c) _ (c x)=P_Z P_θ,t_ (· x)(C=c). The context value Γℓ(c,x)=[hℓ−1(Z)∣C=c] _ (c;x)=E[h_ -1(Z) C=c] averages over the token at J under the same base transition. Below, expectations and variances indexed by νℓ _ are over C∼νℓ(⋅∣x)C _ (· x). Given C=cC=c, the token at J has conditional law pℓh(⋅∣c,x)p_ ^h(· c,x) under the exact Doob transition Pθ,tℓh(⋅∣x)P_θ,t_ ^h(· x). The enumerated tokens form SM(c)=a1,…,aM⊆S_M(c)=\a_1,…,a_M\ . Restricting the exact conditional to this set and renormalizing gives pℓ,Mh(a∣c,x)=pℓh(a∣c,x)a∈SM(c)pℓh(SM(c)∣c,x).p_ ,M^h(a c,x)= p_ ^h(a c,x)1\a∈ S_M(c)\p_ ^h(S_M(c) c,x). Using h^ℓ−1 h_ -1 instead induces the practical conditional p^ℓ,M(⋅∣c,x) p_ ,M(· c,x), averaged over any completion randomness. Combining this conditional with the unchanged context law νℓ(⋅∣x) _ (· x) defines the full-state transition P^θ,tℓ(M)(⋅∣x) P_θ,t_ ^(M)(· x). Table 1: DNA design with hard and soft gates. Activity + ATAC results with held-out activity evaluation. Both modes use the same success criteria, differing only in whether activity receives a hard gate (binary penalty) or a soft gate (smooth reward). Method Joint SR (%) ↑ Activity ↑ ATAC ↑ Uniq. (%) 3-mer r JASPAR r Time (ms) Median Top-10 Median Top-10 Hard-Gate Mode Pretrained 0.00± 0.00 0.18± 0.01 2.41± 0.39 0.03± 0.00 0.65± 0.06 100.00± 0.00 -0.04± 0.01 0.29± 0.01 8.46± 0.23 Best-of-N 2.34± 0.34 1.37± 0.12 5.32± 0.19 0.45± 0.01 0.90± 0.04 100.00± 0.00 0.76± 0.00 0.90± 0.00 271.56± 0.68 TDS 78.80± 6.63 4.80± 0.13 6.19± 0.21 0.84± 0.02 0.97± 0.01 48.12± 1.29 0.92± 0.03 0.91± 0.01 171.39± 0.18 SMC 76.04± 8.17 4.74± 0.11 6.25± 0.24 0.83± 0.03 0.97± 0.01 48.02± 2.62 0.92± 0.04 0.91± 0.01 89.28± 0.10 FK-Steering 4.90± 0.58 0.84± 0.06 5.51± 0.14 0.30± 0.01 0.92± 0.01 100.00± 0.00 0.62± 0.01 0.86± 0.00 111.00± 0.83 SVDD 45.05± 0.97 4.14± 0.03 6.62± 0.04 0.89± 0.01 0.99± 0.00 100.00± 0.00 0.69± 0.00 0.91± 0.00 244.61± 0.53 DiMOS (Ours) 87.55± 0.65 5.34± 0.05 6.98± 0.07 0.99± 0.00 1.00± 0.00 100.00± 0.00 0.86± 0.00 0.93± 0.00 234.63± 0.33 Soft-Gate Mode Pretrained 0.00± 0.00 0.18± 0.01 2.41± 0.39 0.03± 0.00 0.65± 0.06 100.00± 0.00 -0.04± 0.01 0.29± 0.01 8.46± 0.23 Best-of-N 2.08± 0.39 1.88± 0.03 5.29± 0.22 0.42± 0.01 0.90± 0.03 100.00± 0.00 0.79± 0.01 0.91± 0.01 240.96± 0.65 TDS 47.40± 7.16 4.14± 0.15 6.15± 0.33 0.79± 0.02 0.95± 0.01 47.97± 1.55 0.79± 0.02 0.88± 0.02 200.87± 0.93 SMC 41.20± 5.15 4.02± 0.18 5.71± 0.05 0.81± 0.02 0.95± 0.02 48.39± 1.48 0.74± 0.05 0.86± 0.01 89.73± 0.15 FK-Steering 4.84± 0.34 1.45± 0.07 5.76± 0.12 0.25± 0.00 0.92± 0.01 100.00± 0.00 0.61± 0.02 0.85± 0.01 110.62± 0.32 SVDD 55.16± 1.55 4.66± 0.01 7.24± 0.09 0.83± 0.01 0.99± 0.00 100.00± 0.00 0.74± 0.01 0.93± 0.00 244.70± 0.56 DiMOS (Ours) 98.39± 0.29 5.95± 0.02 7.32± 0.06 0.99± 0.00 0.99± 0.00 100.00± 0.00 0.88± 0.00 0.93± 0.00 234.60± 0.57 Theorem 1 (Full-state error of coordinate-wise Doob resampling). For any 1≤M≤||1≤ M≤|V|, suppose Γℓ(C,x)>0 _ (C;x)>0 and pℓh(SM(C)∣C,x)>0p_ ^h(S_M(C) C,x)>0 almost surely under νℓ(⋅∣x) _ (· x). Then hℓ(x)=νℓ[Γℓ(C,x)]h_ (x)=E_ _ [ _ (C;x)] and TV(P^θ,tℓ(M)(⋅∣x),Pθ,tℓh(⋅∣x)) \! ( P_θ,t_ ^(M)(· x),P_θ,t_ ^h(· x) ) ≤12νℓ[|Γℓ(C,x)hℓ(x)−1|]⏟context-reweighting error+νℓ[pℓh(SM(C)c∣C,x)]⏟top-M truncation error ≤ 12E_ _ \! [ | _ (C;x)h_ (x)-1 | ]_context-reweighting error+ E_ _ \! [p_ ^h(S_M(C)^c C,x) ]_top-$M$ truncation error (14) +νℓ[TV(p^ℓ,M(⋅∣C,x),pℓ,Mh(⋅∣C,x))]⏟value-estimation error. + E_ _ \! [TV\! ( p_ ,M(· C,x),p_ ,M^h(· C,x) ) ]_value-estimation error. The first term is at most Varνℓ[Γℓ(C,x)]/(2hℓ(x)) Var_ _ [ _ (C;x)]/(2h_ (x)). The proof and empirical diagnostics are in Appendices B and C. When M=||M=|V|, the truncation term vanishes. If the completion-based values approximate the exact Doob values well, the value-estimation term is also small, leaving the discrepancy primarily controlled by the variation of the shared-context value. 4.3 Budget-Efficient Trajectory Search Jointly satisfying multiple requirements can leave a small feasible region (Xia et al., 2025), making repeated independent sampling costly. We therefore share trajectory prefixes and allocate more search budget to branches with higher estimated joint rewards. Stage-Wise Branching. To control the cost of multi-trajectory exploration, we share a denoising prefix within each of B independent generation groups, each returning one design. Each group follows a single trajectory for s steps, then copies its partially denoised state into K branches. These branches explore alternative continuations through the stochastic local updates of Sec. 4.2. The branch width K determines how many continuations are explored concurrently, while the split step s trades a longer shared prefix for a shorter branching phase. Reward-Guided Refinement. At post-split checkpoints, we score branches using the completion-based estimate of Sec. 4.2. Within each group, we copy the state with the highest estimated joint reward into all K slots, then use stochastic local updates to explore new continuations. Each group returns its highest-reward final design. Group-local selection prevents one trajectory from replacing other groups’ outputs, helping preserve sequence uniqueness. Prefix sharing reduces state updates over L steps from KLKL to s+K(L−s)s+K(L-s) per group. Each guided update compares up to M candidates per branch, with additional reward evaluations for branch selection (Appendix E). 5 Experiments In this section, we evaluate DiMOS on DNA, protein, and RNA design using frozen generative priors. We assess joint success rate (SR) and design quality under hard-gate and soft-gate modes, then examine component contributions, hyperparameter sensitivity, and time-budget trade-offs. 5.1 Experimental Setup Tasks and Gate Modes. We study ❶ Activity + ATAC for DNA regulatory activity and chromatin accessibility (Gosai et al., 2024; Wang et al., 2025), and ❷ ΔΔG G + SeqRec for protein stability and sequence recovery. Both tasks require joint threshold satisfaction. In hard-gate mode, activity or ΔΔG G uses the binary penalty of Eq. (10), while the other objective retains its soft gate. In soft-gate mode, both objectives use soft gates. Success criteria remain unchanged. This tests guidance when one requirement provides no useful gradient. Additional DNA, protein, and RNA results appear in Appendix I. Baselines. We compare sampling and selection (Pretrained, Best-of-N), gradient-based steering (TDS) (Wu et al., 2023; Nisonoff et al., 2025), and gradient-free steering (SMC, FK-Steering, SVDD) (Singhal et al., 2025; Li et al., 2025). All methods share the frozen prior within each task, and most baselines have generation times of the same order of magnitude. Implementation and budget details are in Appendix G. Evaluation Metrics. We report joint SR, the fraction of outputs satisfying both thresholds, and each objective’s median over all outputs and over the ten designs with the highest joint reward. We also report uniqueness (the fraction of distinct sequences), domain-specific naturalness and validity checks, and generation time per design. Tabs. 1 and 2 report mean ± s.d. over three seeds. Full metric definitions and thresholds are in Appendix F. 5.2 Main Results Table 2: Protein design with hard and soft gates. ΔΔG G + SeqRec results with identical success criteria. The ΔΔG G term uses a hard gate (binary penalty) or a soft gate (smooth reward). Method Joint SR (%) ↑ ΔΔG G ↑ SeqRec ↑ Uniq. (%) scRMSD (%) pLDDT (%) Time (ms) Median Top-10 Median Top-10 Hard-Gate Mode Pretrained 4.04± 0.30 -0.45± 0.00 0.45± 0.03 0.44± 0.00 0.48± 0.00 99.67± 0.05 93.55± 0.11 87.98± 0.11 17.42± 0.08 Best-of-N 16.06± 0.19 -0.02± 0.01 0.70± 0.05 0.49± 0.00 0.52± 0.00 99.18± 0.03 92.84± 0.09 86.65± 0.16 159.89± 0.05 TDS 13.59± 0.48 -0.12± 0.01 0.38± 0.06 0.47± 0.00 0.50± 0.00 16.75± 0.24 93.12± 1.00 87.96± 0.21 124.49± 0.38 SMC 4.10± 0.30 -0.10± 0.06 0.69± 0.07 0.44± 0.00 0.48± 0.00 22.09± 0.74 92.90± 0.83 89.37± 0.03 83.11± 0.08 FK-Steering 8.03± 0.08 -0.26± 0.01 0.71± 0.03 0.45± 0.00 0.50± 0.00 99.93± 0.00 94.23± 0.16 87.43± 0.19 192.43± 0.12 SVDD 31.68± 0.03 0.07± 0.01 0.75± 0.02 0.54± 0.00 0.57± 0.00 98.61± 0.12 93.86± 0.13 84.64± 0.09 166.79± 0.09 DiMOS (Ours) 58.38± 0.36 0.07± 0.00 1.38± 0.00 0.61± 0.00 0.63± 0.00 100.00± 0.00 92.32± 0.05 85.92± 0.03 157.63± 0.05 Soft-Gate Mode Pretrained 4.04± 0.30 -0.45± 0.00 0.45± 0.03 0.44± 0.00 0.48± 0.00 99.67± 0.05 93.55± 0.11 87.98± 0.11 17.38± 0.10 Best-of-N 17.77± 0.19 0.01± 0.01 0.73± 0.04 0.49± 0.00 0.52± 0.00 99.00± 0.08 94.21± 0.05 85.81± 0.05 159.68± 0.13 TDS 9.14± 0.45 -0.31± 0.02 0.41± 0.04 0.47± 0.00 0.49± 0.00 17.34± 0.19 94.88± 0.59 90.67± 0.17 123.78± 0.58 SMC 6.79± 0.43 -0.48± 0.05 0.53± 0.03 0.45± 0.00 0.48± 0.00 22.14± 0.97 92.25± 0.37 86.24± 0.21 82.51± 0.03 FK-Steering 6.38± 0.19 -0.33± 0.00 0.44± 0.01 0.46± 0.00 0.50± 0.00 99.54± 0.00 93.84± 0.11 87.63± 0.11 192.32± 0.06 SVDD 35.66± 0.41 0.38± 0.01 1.00± 0.02 0.53± 0.00 0.56± 0.00 96.83± 0.21 96.57± 0.12 83.90± 0.03 166.70± 0.08 DiMOS (Ours) 70.64± 0.24 0.51± 0.01 1.39± 0.03 0.60± 0.00 0.63± 0.00 99.93± 0.00 93.99± 0.21 85.61± 0.09 157.51± 0.11 Tabs. 1 and 2 summarize DNA and protein results under both gate modes. Three findings emerge. ❶ Joint Objective Improvement. DiMOS leads both domains and gate modes in joint SR, including with a non-differentiable hard gate. Its evaluation-only guidance uses reward values without smoothing that gate. In soft-gate mode, it nearly doubles joint SR relative to SVDD, the strongest baseline, while raising both objective medians, showing distribution-wide gains beyond top-ranked designs. ❷ Effective Generation-Time Guidance. At comparable generation times, DiMOS substantially outperforms Best-of-N, which ranks only completed samples. This supports integrating reward feedback into generation rather than relying solely on final selection to find jointly feasible designs within a given budget. ❸ Diverse High-Quality Outputs. DiMOS retains 100%100\% uniqueness on DNA and over 99.9%99.9\% on protein under both gate modes. In DNA hard-gate mode, distinct successful designs account for 87.55%87.55\% of its outputs, versus at most 48.12%48.12\% for TDS, bounded by its uniqueness. This is consistent with restricting branch replication to independent generation groups (Sec. 4.3). DiMOS retains high DNA motif-profile correlations and over 92%92\% protein fold recovery (scRMSD), providing quality checks beyond the optimized objectives. Figure 3: Sensitivity to M, K, and s/Ts/T. One parameter varies at a time. Configuration DNA Protein Hard Gate Soft Gate Hard Gate Soft Gate Full DiMOS 87.55± 0.65 98.39± 0.29 58.38± 0.36 70.64± 0.24 Doob-guided local resampling – Doob guidance -62.92± 1.84 -64.95± 0.46 -46.96± 0.24 -56.18± 0.35 – enumeration -12.66± 1.72 -5.26± 0.22 -38.28± 0.08 -47.61± 0.08 Budget-efficient trajectory search – stage-wise branching -51.61± 2.03 -34.69± 3.29 -6.47± 0.19 -7.75± 0.23 – reward-guided refinement -18.39± 1.63 -7.03± 1.10 -0.11± 0.27 -1.80± 0.08 Table 3: Component ablation. Each row removes one component; entries are the change in joint SR (%) from the full configuration, over three seeds. Figure 4: Joint success versus generation time. DiMOS budget envelopes on Activity + ATAC (DNA) and ΔΔG G + SeqRec (protein). 5.3 Hyperparameter Analysis and Ablation Hyperparameter Sensitivity. Fig. 3 examines candidate width M, branch width K, and normalized search onset s/Ts/T on DNA and protein. Increasing M and K generally improves joint SR in both gate modes, although gains diminish as more branches are added. Search onset trades prefix sharing against multi-trajectory exploration. DNA maintains high success when branching is delayed through roughly the first half of denoising. A later split propagates one shared trajectory instead of K branches for more steps, allowing DNA to save computation on the prefix with little loss in success. Protein benefits from earlier branching, favoring more computation on alternative continuations rather than a long shared prefix. The allocation should therefore reflect the task. Component Contributions. Tab. 3 tests what each component adds beyond the remaining algorithm. ❶ Local Reward Feedback. Removing Doob guidance disables the reward term in local resampling but retains candidate priors and reward-based branch selection. This causes the largest drop in joint SR, showing that trajectory-level selection alone cannot replace reward-informed token updates. ❷ Candidate Construction. Replacing enumeration with M i.i.d. token draws at the same selected position allows repeats while keeping the candidate count and reweighting rule unchanged. The larger losses on protein highlight the value of evaluating distinct alternatives, not merely more candidates. ❸ Multiple Continuations. Setting K=1K=1 retains local guidance but collapses search to a single trajectory. The larger losses on DNA highlight the value of exploring multiple continuations rather than committing to one guided path. ❹ Intermediate Selection. Disabling refinement retains the K branches and final best-of-K selection but removes intermediate pruning and replication. The losses, especially on DNA, show that multiple trajectories benefit from redirecting computation toward promising partial designs during search. App. H transplants this search layer onto the SVDD kernel at matched wall clock: it improves that kernel too, but closes under a fifth of the gap to DiMOS, leaving the local transition as the larger factor. 5.4 Efficiency under Time Budgets Fig. 4 plots an empirical Pareto frontier between generation time and joint SR from single-seed, soft-gate sweeps. The staircase gives the highest observed joint SR within each time budget, revealing two benefits. ❶ Higher Success at a Given Budget. At the time budgets of the compared reward-based baselines, DiMOS achieves higher joint SR in both domains. It therefore uses the available generation budget to produce a higher proportion of jointly feasible designs. ❷ Lower Cost at a Given Success Rate. Measured DiMOS configurations match or exceed SVDD’s joint SR with approximately 63%63\% less generation time on DNA and 74%74\% less on protein. Practitioners can therefore choose higher success within a fixed budget or lower cost for a target success rate. 6 Conclusion We introduced DiMOS, a training-free framework for multi-objective scientific design with frozen masked diffusion models. It combines approximate Doob-guided local resampling with budget-efficient trajectory search, accommodating hard constraints and black-box rewards without gradients. Experiments across DNA, protein, and RNA show improved joint success while retaining diverse designs. Allocation studies further highlight task-dependent trade-offs between candidate width, branching, and search duration. Acknowledgements Han Liu is partially supported by NIH R01LM1372201, NSF AST-2421845, Simons Foundation MPS-AI-00010513, AbbVie, Dolby and Chan Zuckerberg Biohub Chicago Spoke Award. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies. The authors declare no competing interests. References Austin et al. (2021) J. Austin, D. D. Johnson, J. Ho, D. Tarlow, and R. van den Berg Structured denoising diffusion models in discrete state-spaces. In Advances in Neural Information Processing Systems, Vol. 34, p. 17981–17993. External Links: Link Cited by: §3. Black et al. (2024) K. Black, M. Janner, Y. 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By the Markov property of the original reverse process and the definition of hℓh_ , for any state x at time tℓt_ , hℓ(x) h_ (x) =ℙθ[ϕ(X0)∣Xtℓ=x] =E_P_θ [φ(X_0) X_t_ =x ] =ℙθ[hℓ−1(Xtℓ−1)∣Xtℓ=x] =E_P_θ [h_ -1(X_t_ -1) X_t_ =x ] =∑yPθ,tℓ(y∣x)hℓ−1(y). = _yP_θ,t_ (y x)h_ -1(y). (15) Since ϕφ is strictly positive, so is hℓh_ on the states reached by the reverse sampler. Therefore, ∑yPθ,tℓh(y∣x) _yP_θ,t_ ^h(y x) =1hℓ(x)∑yPθ,tℓ(y∣x)hℓ−1(y)=1. = 1h_ (x) _yP_θ,t_ (y x)h_ -1(y)=1. (16) Moreover, Pθ,tℓh(y∣x)≥0P_θ,t_ ^h(y x)≥ 0, so Pθ,tℓh(⋅∣x)P_θ,t_ ^h(· x) is a valid transition kernel. Let ℙθhP_θ^h denote the path law induced by the transformed kernels. For a trajectory xL,xL−1,…,x0x_L,x_L-1,…,x_0 starting from the fixed state xLx_L, ℙθh(XtL−1=xL−1,…,Xt0=x0∣XtL=xL) _θ^h (X_t_L-1=x_L-1,…,X_t_0=x_0 X_t_L=x_L ) =∏ℓ=1LPθ,tℓh(xℓ−1∣xℓ) = _ =1^LP_θ,t_ ^h(x_ -1 x_ ) =∏ℓ=1LPθ,tℓ(xℓ−1∣xℓ)hℓ−1(xℓ−1)hℓ(xℓ) = _ =1^LP_θ,t_ (x_ -1 x_ ) h_ -1(x_ -1)h_ (x_ ) =[∏ℓ=1LPθ,tℓ(xℓ−1∣xℓ)]ϕ(x0)hL(xL), = [ _ =1^LP_θ,t_ (x_ -1 x_ ) ] φ(x_0)h_L(x_L), (17) where the h-ratios telescope and h0(x0)=ϕ(x0)h_0(x_0)=φ(x_0). Summing over x1,…,xL−1x_1,…,x_L-1 gives ℙθh(Xt0=x0∣XtL=xL) _θ^h (X_t_0=x_0 X_t_L=x_L ) =ϕ(x0)hL(xL)ℙθ(Xt0=x0∣XtL=xL) = φ(x_0)h_L(x_L)P_θ (X_t_0=x_0 X_t_L=x_L ) =pθ(x0)ϕ(x0)hL(xL). = p_θ(x_0)φ(x_0)h_L(x_L). (18) By definition of the h-function and because the original sampler starts from the fixed state xLx_L, hL(xL) h_L(x_L) =ℙθ[ϕ(X0)∣XtL=xL] =E_P_θ [φ(X_0) X_t_L=x_L ] =∑x0pθ(x0)ϕ(x0)=X0∼pθ[ϕ(X0)]. = _x_0p_θ(x_0)φ(x_0)=E_X_0 p_θ[φ(X_0)]. (19) Substituting this into Eq. (18) yields ℙθh(Xt0=x0∣XtL=xL)=pθ(x0)ϕ(x0)X0∼pθ[ϕ(X0)]=πϕ(x0).P_θ^h (X_t_0=x_0 X_t_L=x_L )= p_θ(x_0)φ(x_0)E_X_0 p_θ[φ(X_0)]= _φ(x_0). Hence the transformed chain has terminal marginal πϕ _φ. ∎ Appendix B Proof of Theorem 1 Proof. Fix x=xtℓx=x_t_ and suppress the dependence on x and tℓt_ when there is no ambiguity. Let A=ZJA=Z^J denote the token at the selected coordinate J. For a context c=(e,b)c=(e,b) and token a∈a , define yx(c,a)y_x(c,a) as the unique successor state whose reveal set is e, whose token at J is a, and whose other newly revealed tokens are given by b; all remaining coordinates agree with x. Define the corresponding base conditional by pℓ(a∣c,x)=ℙ(A=a∣C=c,Xtℓ=x).p_ (a c,x)=P(A=a C=c,X_t_ =x). For fixed x, the correspondence between the successor Z and the pair (C,A)(C,A) is one-to-one: Z determines its reveal set E, the selected coordinate J, and hence (C,A)(C,A), while (C,A)(C,A) reconstructs Z as yx(C,A)y_x(C,A). Therefore, Pθ,tℓ(yx(c,a)∣x)=νℓ(c∣x)pℓ(a∣c,x).P_θ,t_ \! (y_x(c,a) x )= _ (c x)\,p_ (a c,x). (20) By the definition of the context value, Γℓ(c,x)=∑a∈pℓ(a∣c,x)hℓ−1(yx(c,a)). _ (c;x)= _a p_ (a c,x)\,h_ -1\! (y_x(c,a) ). Therefore, using the backward recursion for the Doob value, hℓ(x) h_ (x) =∑yPθ,tℓ(y∣x)hℓ−1(y) = _yP_θ,t_ (y x)\,h_ -1(y) =∑cνℓ(c∣x)∑a∈pℓ(a∣c,x)hℓ−1(yx(c,a)) = _c _ (c x) _a p_ (a c,x)\,h_ -1\! (y_x(c,a) ) =∑cνℓ(c∣x)Γℓ(c,x)=νℓ[Γℓ(C,x)]. = _c _ (c x)\, _ (c;x)=E_ _ [ _ (C;x) ]. (21) We next characterize the exact Doob transition. From Eq. (20), Pθ,tℓh(yx(c,a)∣x) P_θ,t_ ^h\! (y_x(c,a) x ) =Pθ,tℓ(yx(c,a)∣x)hℓ−1(yx(c,a))hℓ(x) =P_θ,t_ \! (y_x(c,a) x ) h_ -1(y_x(c,a))h_ (x) =νℓ(c∣x)Γℓ(c,x)hℓ(x)pℓh(a∣c,x), = _ (c x) _ (c;x)h_ (x)p_ ^h(a c,x), (22) where pℓh(a∣c,x)=pℓ(a∣c,x)hℓ−1(yx(c,a))Γℓ(c,x).p_ ^h(a c,x)= p_ (a c,x)h_ -1(y_x(c,a)) _ (c;x). (23) Thus the exact Doob transition changes both the conditional law of the token at J and the marginal law of the shared context. In particular, the latter becomes νℓh(c∣x)=νℓ(c∣x)Γℓ(c,x)hℓ(x). _ ^h(c x)= _ (c x) _ (c;x)h_ (x). (24) To separate the three approximation sources, introduce two intermediate coordinate-wise kernels. First, let P~θ,tℓh P_θ,t_ ^h keep C∼νℓ(⋅∣x)C _ (· x) unchanged and sample the token at J from the exact conditional pℓhp_ ^h: P~θ,tℓh(yx(c,a)∣x)=νℓ(c∣x)pℓh(a∣c,x). P_θ,t_ ^h\! (y_x(c,a) x )= _ (c x)\,p_ ^h(a c,x). (25) Second, let P~θ,tℓh,(M) P_θ,t_ ^h,(M) additionally restrict the token to the enumerated set SM(c)S_M(c): P~θ,tℓh,(M)(yx(c,a)∣x)=νℓ(c∣x)pℓ,Mh(a∣c,x). P_θ,t_ ^h,(M)\! (y_x(c,a) x )= _ (c x)\,p_ ,M^h(a c,x). (26) Finally, by definition of the practical resampling rule, P^θ,tℓ(M)(yx(c,a)∣x)=νℓ(c∣x)p^ℓ,M(a∣c,x), P_θ,t_ ^(M)\! (y_x(c,a) x )= _ (c x)\, p_ ,M(a c,x), (27) where p^ℓ,M p_ ,M includes the completion-based value approximation, averaged over any completion randomness. We now bound the discrepancy between consecutive kernels. Context-Reweighting Error. The kernels P~θ,tℓh P_θ,t_ ^h and Pθ,tℓhP_θ,t_ ^h have the same conditional law pℓh(⋅∣c,x)p_ ^h(· c,x) given C=cC=c and differ only in their context marginals. Using Eqs. (22) and (25), TV(P~θ,tℓh(⋅∣x),Pθ,tℓh(⋅∣x)) \! ( P_θ,t_ ^h(· x),P_θ,t_ ^h(· x) ) =12∑c,aνℓ(c∣x)|1−Γℓ(c,x)hℓ(x)|pℓh(a∣c,x) = 12 _c,a _ (c x) |1- _ (c;x)h_ (x) |p_ ^h(a c,x) =12νℓ[|Γℓ(C,x)hℓ(x)−1|]. = 12E_ _ \! [ | _ (C;x)h_ (x)-1 | ]. (28) Top-M Truncation Error. For each context c, pℓ,Mhp_ ,M^h is obtained by restricting pℓhp_ ^h to SM(c)S_M(c) and renormalizing. Hence TV(pℓ,Mh(⋅∣c,x),pℓh(⋅∣c,x))=pℓh(SM(c)c∣c,x).TV\! (p_ ,M^h(· c,x),p_ ^h(· c,x) )=p_ ^h(S_M(c)^c c,x). (29) Indeed, writing qc=pℓh(SM(c)∣c,x)q_c=p_ ^h(S_M(c) c,x), 12∑a∈|pℓ,Mh(a∣c,x)−pℓh(a∣c,x)| 12 _a |p_ ,M^h(a c,x)-p_ ^h(a c,x) | =12[∑a∈SM(c)pℓh(a∣c,x)(1qc−1)+∑a∉SM(c)pℓh(a∣c,x)] = 12 [ _a∈ S_M(c)p_ ^h(a c,x) ( 1q_c-1 )+ _a∉ S_M(c)p_ ^h(a c,x) ] =12[(1−qc)+(1−qc)]=1−qc. = 12 [(1-q_c)+(1-q_c) ]=1-q_c. Since the two full-state kernels share the same context law νℓ _ , Eq. (29) gives TV(P~θ,tℓh,(M)(⋅∣x),P~θ,tℓh(⋅∣x)) \! ( P_θ,t_ ^h,(M)(· x), P_θ,t_ ^h(· x) ) =νℓ[pℓh(SM(C)c∣C,x)]. =E_ _ \! [p_ ^h(S_M(C)^c C,x) ]. (30) Value-Estimation Error. The practical and truncated-oracle kernels also share the same context law νℓ _ . Therefore, TV(P^θ,tℓ(M)(⋅∣x),P~θ,tℓh,(M)(⋅∣x)) \! ( P_θ,t_ ^(M)(· x), P_θ,t_ ^h,(M)(· x) ) =12∑cνℓ(c∣x)∑a∈|p^ℓ,M(a∣c,x)−pℓ,Mh(a∣c,x)| = 12 _c _ (c x) _a | p_ ,M(a c,x)-p_ ,M^h(a c,x) | =νℓ[TV(p^ℓ,M(⋅∣C,x),pℓ,Mh(⋅∣C,x))]. =E_ _ \! [TV\! ( p_ ,M(· C,x),p_ ,M^h(· C,x) ) ]. (31) Applying the triangle inequality along P^θ,tℓ(M)⟶P~θ,tℓh,(M)⟶P~θ,tℓh⟶Pθ,tℓh P_θ,t_ ^(M)\; \; P_θ,t_ ^h,(M)\; \; P_θ,t_ ^h\; \;P_θ,t_ ^h and substituting Eqs. (28), (30), and (31) yields Eq. (14). Finally, Eq. (21) implies νℓ[Γℓ(C,x)]=hℓ(x)E_ _ [ _ (C;x)]=h_ (x). By Cauchy–Schwarz, 12νℓ[|Γℓ(C,x)hℓ(x)−1|] 12E_ _ \! [ | _ (C;x)h_ (x)-1 | ] =νℓ[|Γℓ(C,x)−hℓ(x)|]2hℓ(x) = E_ _ [| _ (C;x)-h_ (x)|]2h_ (x) ≤Varνℓ[Γℓ(C,x)]2hℓ(x). ≤ Var_ _ [ _ (C;x)]2h_ (x). (32) This proves the claimed bound. ∎ The branch-level pruning of Sec. 4.3 sits outside this bound: it serves search rather than exact sampling from the terminal-reweighted distribution. Appendix C Empirical Evaluation of Local Approximation Error We evaluate the single-step approximation in Theorem 1 on DNA Activity + ATAC, protein ΔΔG G + sequence recovery, and RNA MRL + accessibility. For each domain and gate mode, we collect 2424 partially generated sequences from three seeds and eight generation groups per seed. Each group contributes its first branch when three MASK positions remain, using the main-table configurations. Protein samples are balanced across the first six held-out backbones reaching this count. This choice makes the reference calculation tractable. Exact h-values average terminal exponential rewards over all remaining base-process continuations, whose number grows rapidly with the number of MASK positions. Earlier in decoding, Monte Carlo estimation is more affordable but introduces sampling uncertainty, especially when a few continuations dominate the reward weights. Using three MASK positions permits exact enumeration across all three domains, isolating the sampler’s approximation error from Monte Carlo reference error at a common near-terminal setting. We compute the reference Doob transition by enumerating the remaining base-process continuations and compare it with the practical local resampling kernel. The reference includes all token classes, while practical menus and completion rules remain unchanged. Completion randomness is integrated where applicable, so the reported TV measures the full single-step discrepancy rather than only the context term. Production-code replays verify the practical resampling probabilities. Table 4: Local approximation error across domains. Each row summarizes 2424 partially generated sequences with three remaining MASK positions. Domain Gate Samples Median TV Mean TV DNA Soft 24 0.00041 0.00286 DNA Hard 24 0.00012 0.00968 Protein Soft 24 0.00073 0.09231 Protein Hard 24 0.00031 0.03408 RNA Soft 24 0.02280 0.03571 RNA Hard 24 0.00123 0.01503 Tab. 4 shows median TV below 0.0010.001 for DNA and protein under both gate modes and below 0.0230.023 for RNA, indicating close agreement for typical sampled updates in this near-terminal regime. Appendix D Why the Hard-Gate Penalty Is Finite The log-space aggregation in Eq. (10) gives the terminal weight exp(R(x)/β)=∏k=1n[exp(gk(dk(x)))]wk/β. (R(x)/β )= _k=1^n [ (g_k(d_k(x)) ) ]^w_k/β. (33) A soft gate contributes σ(dk(x)/τk)wk/βσ(d_k(x)/ _k)^w_k/β, while a hard gate contributes 11 when satisfied and e−Pwk/βe^-Pw_k/β otherwise. Thus a finite hard penalty discourages violations without excluding them from the support. Two properties of this construction shape the sampler. A Constant Penalty Cancels. When all candidate completions violate the same hard-gated requirement, its common contribution −Pwk/β-Pw_k/β cancels during local normalization. Relative weights then depend on the prior and the remaining requirements. A hard gate distinguishes candidates on opposite sides of its threshold, but provides no ranking on either side. A soft gate additionally distinguishes their margins. The Penalty Is Finite. A finite penalty keeps local resampling well-defined when all candidate completions violate at least one hard-gated requirement, allowing the remaining reward feedback to guide selection. In the limit P→∞P→∞, these candidates receive zero weight and the local weights cannot be normalized. The strength of violation suppression depends on Pwk/βPw_k/β relative to the other contributions to the log weight, rather than on P alone. Appendix E Algorithm and Implementation Details In this section, we detail the implementation of DiMOS. We first specify the sampling schedule and transition notation, then present the full algorithm and explain how completions are used for local guidance and branch scoring. E.1 Sampling Schedule and Transition Notation Algorithm 1 describes local enumeration and branch search for the main-text experiments. We index denoising steps by i=0,…,L−1i=0,…,L-1, so branching occurs before the first update when s=0s=0. The corresponding noise index is ℓ=L−i =L-i. Token enumeration is enabled from the first guided update, independently of the split step. The configured guidance window is G; branch-selection checkpoints are =i:s<i<L−1,(i−s)modf=0C=\i:s<i<L-1,\ (i-s) f=0\, where f is the search interval. The final update is unguided in the default configurations. Reveal returns a shared base successor z and its newly revealed positions E: it chooses a count using the configured reveal schedule, selects that many remaining masked positions uniformly, and samples their tokens. Previously committed tokens stay fixed. We write qijq_i^j for the actual conditional token distribution after proposal tempering, and QiQ_i for the corresponding IID transition. For enumeration, all candidates share z except at the selected position, so only the factor qij(am)q_i^j(a_m) varies in their prior weights. When E is empty, the code uses IID candidates from QiQ_i and reward-only weights, without multiplying their proposal probability a second time. Table 5: Sampler configuration of the reported DiMOS runs, one row per task. T is the number of executed denoising steps, M the candidate width, K the branch width, s the split step, β the tilt temperature of Eq. (8), and η the proposal temperature. Domain Task T M K s β η DNA ATAC 128128 44 44 6464 0.250.25 3.03.0 MaxGap 128128 44 88 6464 0.250.25 3.03.0 Protein SeqRec 4949 44 22 00 0.020.02 1.01.0 GRAVY 4949 22 22 00 0.020.02 1.01.0 RNA Accessibility 4949 44 22 1818 0.040.04 0.050.05 MFE 4949 44 22 1818 0.040.04 1.01.0 E.2 DIMOS Algorithm Algorithm 1 DiMOS with local token enumeration and branch search 1: frozen denoiser μθ _θ; joint reward R; B lineages, L steps, M tokens, K branches; split s; guidance steps G, search checkpoints C; temperatures η,βη,β 2: B clean designs 3: initialize one fully masked state in each of B independent lineages 4: for i=0,…,L−1i=0,…,L-1 do 5: ℓ←L−i ← L-i 6: if i=si=s and K>1K>1 then 7: clone each lineage’s state into K branches 8: end if 9: for all active states x in parallel do 10: if i∈i and x contains masks then 11: compute tempered token conditionals qiq_i from μθ(x,tℓ) _θ(x,t_ ) using η 12: (z,E)←Reveal(x,qi,i)(z,E)← Reveal(x,q_i,i) 13: if E≠∅E≠ then ⊳ enumerated menu 14: j←argmaxu∈EH(qiu)j← _u∈ EH(q_i^u) 15: (a1,…,aM)←TopM(qij)(a_1,…,a_M) (q_i^j) ⊳ distinct tokens 16: y(m)←zj←amy^(m)← z^j← a_m, bm←logqij(am)b_m← q_i^j(a_m) for each m 17: else⊳ empty-reveal fallback 18: draw y(1),…,y(M)y^(1),…,y^(M) independently from Qi(⋅∣x)Q_i(· x); bm←0b_m← 0 19: end if 20: x^0(m)←Complete(y(m),tℓ−1) x^(m)_0← Complete(y^(m),t_ -1); R(m)←R(x^0(m))R^(m)← R( x^(m)_0) 21: logw^m←bm+R(m)/β w_m← b_m+R^(m)/β 22: m⋆∼Categorical(softmax(logw^1:M))m (softmax( w_1:M)); x←y(m⋆)x← y^(m ) 23: else 24: advance x with the unguided reverse update 25: end if 26: end for 27: if K>1K>1 and i∈i then 28: compute each branch’s reward as R(Completeargmax(x))R( Complete_argmax(x)) 29: within each lineage, copy the highest-reward branch into all K slots 30: end if 31: end for 32: apply the configured final denoising to any residual masks 33: return the highest-reward clean branch from each lineage E.3 Completion and Branch Reward Estimation Complete preserves committed tokens and fills only remaining masks using the configured completion rule: stochastic completion with shared random numbers across a parent’s candidates for DNA, and argmax completion for protein/RNA. Denoiser outputs are reused for branch reward estimation where available; a reward evaluation is reused only when the completion is the same. Branch selection uses an interval of f=1f=1. Appendix F Evaluation Criteria In this section, we define the task-specific success thresholds and the metrics used to assess joint feasibility, design quality, diversity, validity, and efficiency. F.1 Task Success Thresholds Tab. 6 lists the final evaluation criteria, shared across methods and gate modes within each task. Table 6: Canonical joint-success criteria for the six design tasks. Both objective conditions must be satisfied. Task Objective 1 Objective 2 Activity + ATAC Activity ≥4.0≥ 4.0 ATAC ≥0.7≥ 0.7 Activity + MaxGap Activity ≥4.0≥ 4.0 MaxGap ≥1.0≥ 1.0 ddG + SeqRec ddG ≥0≥ 0 SeqRec ≥0.55≥ 0.55 ddG + GRAVY ddG ≥0≥ 0 GRAVY ≤−0.70≤-0.70 MRL + Opening dG Opening dG ≤0.25≤ 0.25 kcal/mol Optimus-MRL ≥0.45≥ 0.45 MRL + MFE MFE ≤−16≤-16 kcal/mol Optimus-MRL ≥0.45≥ 0.45 F.2 Metric Definitions ❶Joint Success Rate (SR) is the primary metric: the fraction of returned designs satisfying both canonical objective thresholds, rather than optimizing one objective at the expense of the other. For RNA, joint SR uses the held-out Optimus-MRL reward model (Sample et al., 2019) together with the task-specific structural criterion, while generation is guided by ConvGRU-MRL. ❷Objective Quality reports each objective’s median over all outputs and its median among the ten designs with the highest joint reward, capturing both distribution-wide steering and best-design quality. Top-10 designs are ranked by the joint reward rather than either objective alone. ❸Diversity and Validity include exact-sequence uniqueness, DNA naturalness, and protein fold recovery. Uniqueness is the percentage of distinct exact sequences. DNA validity uses 3-mer and JASPAR 2022 motif-profile correlations (Castro-Mondragon et al., 2022) to the top-0.1%0.1\% HepG2 reference set. Protein validity uses held-out ESMFold (Lin et al., 2023), reporting the fractions with scRMSD <2<2Å and pLDDT >80>80. These checks expose particle collapse and reward hacking beyond the guidance reward models. ❹Efficiency is generation time per returned design, reported in milliseconds and not ranked. Within any single table all timings come from one measurement batch run with a single process per GPU and with kernel compilation disabled, so that wall clock is not distorted by contention and so that a speed setting never changes a reported result. For Best-of-N, we select the tested candidate count whose measured generation time is closest to DiMOS’s within each task and gate mode. All main-table entries are mean ± population standard deviation over three independent seeds. Appendix G Baseline and Budget Protocol We use the pretrained DNA and structure-conditioned protein generators released with DRAKES (Wang et al., 2025). DNA designs are 200-bp sequences in the GOSAI setting (Gosai et al., 2024). Protein generation is conditioned on 12 held-out backbones. Pretrained samples directly from the frozen prior. Best-of-N draws independent samples and selects by terminal joint reward; its candidate count is calibrated to the closest DiMOS wall-clock budget. TDS combines a gradient-guided proposal with sequential importance correction. SMC uses the pretrained reverse kernel as its proposal and resamples particles; FK-Steering uses the published discrete Feynman–Kac difference potential; and SVDD-PM greedily selects local candidates using posterior-mean values. RNA excludes TDS because neither selected task supplies a differentiable full joint reward. For the gated-reward comparisons, DiMOS and all reward-based baselines share the same reward parameters within each task and gate mode: P=30P=30, wk=1w_k=1 for every requirement, and the soft-gate widths in Tab. 7. In soft-gate mode, both requirements use soft gates. In hard-gate mode, Activity (DNA), ΔΔG G (protein), or MRL (RNA) uses the binary 0/−P0/-P gate, while the other requirement retains its soft gate. The width τk _k is unused for a hard gate. Pretrained does not use rewards during generation. Table 7: Shared soft-gate widths. The widths follow the reward order shown. Domain Reward Pair τ1 _1 τ2 _2 DNA Activity + ATAC 0.50.5 0.050.05 Activity + MaxGap 0.50.5 0.50.5 Protein ΔΔG G + SeqRec 0.30.3 0.030.03 ΔΔG G + GRAVY 0.30.3 0.100.10 RNA MRL + Opening ΔG G 0.050.05 0.250.25 MRL + MFE 0.050.05 1.01.0 Every baseline runs at the defaults published with its own implementation. Best-of-N and SVDD expose a compute budget through their candidate counts K and M, which we set per block to the closest match to the DiMOS generation budget that is not cheaper than it. Every method returns the same number of final designs within each domain: N=640N=640 DNA sequences per seed, N=128N=128 protein sequences per wild-type backbone, and N=128N=128 RNA sequences per seed. Task definitions, frozen priors, and reward models used for evaluation are fixed across methods. DNA activity and protein ddG rewards are computed using model checkpoints separate from those used for guidance. Recorded generation wall time is normalized by the number of returned designs. Appendix H Separating the Search Layer from the Local Kernel Tab. 3 removes components from DiMOS. This section runs the complementary test: it gives the search layer of Sec. 4.3 to a different local kernel. We keep SVDD-PM at its released hyperparameters and change only where computation is spent. Branches are scored with SVDD-PM’s own posterior-mean completion, so no component of Sec. 4.2 enters. Solving the search arm’s cost for the DiMOS oracle budget gives M=4M=4, which is also DiMOS’s own candidate width, so the matched row and DiMOS differ only in the per-step transition. Table 8: Separating the search layer from the local kernel. DNA activity–ATAC, N=640N=640, three seeds. Rows marked Tab. 1 are reproduced from the main results. Oracle counts reward-model forward passes per seed; time is generation wall clock per returned design. Method M K s Joint SR (%) ↑ 3-mer r Oracle Time (ms) Soft-Gate Mode SVDD-PM (Tab. 1) 1212 11 – 55.16± 1.55 0.740.74 1.971.97M 244.7244.7 ++ search, matched budget 44 44 6464 63.02± 2.20 0.730.73 1.961.96M 251.0251.0 ++ search, released M 2020 44 6464 84.43± 0.96 0.790.79 8.518.51M 1101.81101.8 DiMOS (Tab. 1) 44 44 6464 98.39± 0.29 0.880.88 1.941.94M 234.6234.6 Hard-Gate Mode SVDD-PM (Tab. 1) 1212 11 – 45.05± 0.97 0.690.69 1.971.97M 244.6244.6 ++ search, matched budget 44 44 6464 52.34± 0.80 0.650.65 1.961.96M 250.7250.7 ++ search, released M 2020 44 6464 72.86± 1.41 0.710.71 8.518.51M 1101.81101.8 DiMOS (Tab. 1) 44 44 6464 87.55± 0.65 0.860.86 1.941.94M 234.6234.6 The search layer transfers. At 1.03×1.03× the SVDD-PM generation time and a slightly smaller oracle budget, it adds 7.867.86 and 7.297.29 points on the soft and hard gates. The local kernel accounts for the larger share. The matched row and DiMOS share M, K, and s and differ only in how one step selects its next state, and that difference is 35.3735.37 and 35.2135.21 points. Raising the SVDD candidate width until the search arm spends 4.39×4.39× the DiMOS oracle budget and 4.70×4.70× its time still leaves it 13.9613.96 and 14.6914.69 points behind. Naturalness moves the same way: the transferred search costs the SVDD kernel some 3-mer correlation (0.740.74 to 0.730.73 soft, 0.690.69 to 0.650.65 hard), while DiMOS attains the highest value in both modes. Tab. 9 places these arms against the only budget knob the SVDD kernel has on its own. Joint success rises monotonically with the candidate width M, but the marginal return falls from 5.05.0 points per unit M between M=4M=4 and M=8M=8 to 0.90.9 between M=16M=16 and M=20M=20, so buying joint feasibility through candidates alone becomes expensive well before the gap to DiMOS closes. The matched search arm comes within 0.70.7 points of the M=20M=20 row on the soft gate while spending 0.60×0.60× its oracle budget and 0.58×0.58× its time, and lands near M=17M=17 on the hard gate. Reallocating a fixed budget is therefore worth more to this kernel than widening its candidate set. Table 9: SVDD-PM candidate width on DNA activity–ATAC. T=128T=128, N=640N=640, three seeds, no search layer. Oracle and time are shared by the two gate modes to within 0.2%0.2\%. M Oracle Time (ms) Joint SR (%) ↑ 3-mer r Soft Hard Soft Hard 44 0.660.66M 8484 26.30± 0.83 18.59± 1.38 0.690.69 0.640.64 88 1.311.31M 159159 46.25± 1.00 37.71± 1.36 0.720.72 0.680.68 1010 1.641.64M 184184 51.51± 1.67 43.07± 0.70 0.730.73 0.690.69 1212 (Tab. 1) 1.971.97M 245245 55.16± 1.55 45.05± 0.97 0.740.74 0.690.69 1616 2.622.62M 336336 60.21± 2.78 51.56± 1.67 0.750.75 0.710.71 2020 3.283.28M 434434 63.70± 1.09 55.16± 1.85 0.760.76 0.710.71 Appendix I Additional Tasks In this section, we extend the evaluation to DNA cell-type specificity, protein hydrophilicity, and RNA translation paired with accessibility or folding stability. We examine these additional requirements under both gate modes. I.1 DNA Cell-Type Specificity Tab. 10 reports the second DNA task, Activity + MaxGap, in which HepG2 activity is paired with a specificity margin against K562 and SK-N-SH. Unlike Activity + ATAC, the two objectives are in tension: in the reference enhancer set activity and specificity are negatively correlated, so the direction that raises expression in HepG2 tends to raise it in the off-target lines as well. The hard-gate mode replaces activity’s smooth margin feedback with a binary penalty, removing information about how far a candidate lies from the activity threshold. Table 10: DNA cell-type specificity: Activity ++ MaxGap. HepG2 activity paired with a specificity margin against K562 and SK-N-SH. Unlike Activity ++ ATAC the two objectives are in tension. The activity term uses a hard gate or a soft gate, while MaxGap retains its soft gate. With a hard gate, activity’s own pass rate falls and the designs drift toward uniform composition. Uniqueness, 3-mer r and JASPAR r are guardrails, not objectives, and are reported without ranking. N=640N=640 per seed, mean ± s.d. over three seeds; time is ms per design. Method Joint SR (%) ↑ Activity ↑ MaxGap ↑ Uniq. (%) 3-mer r JASPAR r Time (ms) Median Top-10 Median Top-10 Hard-Gate Mode Pretrained 0.00± 0.00 0.18± 0.01 2.84± 0.25 -0.12± 0.00 0.39± 0.30 100.00± 0.00 -0.04± 0.01 0.29± 0.01 8.48± 0.20 Best-of-N 1.20± 0.27 1.16± 0.05 4.49± 0.24 0.44± 0.00 2.37± 0.65 100.00± 0.00 0.40± 0.01 0.74± 0.00 273.38± 0.32 TDS 7.71± 4.23 4.37± 0.10 5.23± 0.20 -0.37± 0.20 1.88± 0.45 46.30± 1.62 0.77± 0.06 0.85± 0.04 171.94± 0.70 SMC 5.89± 1.42 4.35± 0.10 4.79± 0.13 -0.44± 0.07 2.11± 0.49 47.34± 3.77 0.79± 0.03 0.89± 0.01 75.43± 0.19 FK-Steering 1.72± 0.34 0.59± 0.02 4.61± 0.11 0.19± 0.01 3.02± 0.47 100.00± 0.00 0.18± 0.04 0.52± 0.03 69.84± 0.61 SVDD 35.16± 1.60 3.71± 0.06 6.55± 0.09 1.62± 0.07 5.59± 0.35 100.00± 0.00 0.46± 0.01 0.74± 0.01 280.22± 0.44 DiMOS (Ours) 57.03± 0.92 4.34± 0.02 7.54± 0.07 3.11± 0.12 6.68± 0.29 100.00± 0.00 0.26± 0.01 0.74± 0.01 275.35± 0.07 Soft-Gate Mode Pretrained 0.00± 0.00 0.18± 0.01 2.84± 0.25 -0.12± 0.00 0.39± 0.30 100.00± 0.00 -0.04± 0.01 0.29± 0.01 8.48± 0.20 Best-of-N 1.25± 0.22 2.40± 0.03 4.49± 0.23 0.07± 0.04 2.37± 0.65 100.00± 0.00 0.49± 0.03 0.78± 0.02 274.49± 1.16 TDS 9.79± 1.96 2.95± 0.03 5.40± 0.67 0.21± 0.24 3.32± 0.33 46.98± 0.70 0.28± 0.10 0.42± 0.12 167.61± 1.22 SMC 11.09± 5.18 2.71± 0.55 5.34± 0.29 0.21± 0.12 3.08± 0.41 48.80± 0.48 0.34± 0.24 0.53± 0.19 75.09± 0.22 FK-Steering 1.72± 0.46 1.63± 0.11 4.62± 0.14 -0.01± 0.02 2.96± 0.49 100.00± 0.00 0.39± 0.04 0.71± 0.03 69.80± 0.26 SVDD 44.06± 2.13 4.05± 0.04 7.10± 0.13 1.56± 0.07 4.83± 0.39 100.00± 0.00 0.60± 0.00 0.84± 0.00 280.49± 0.39 DiMOS (Ours) 72.60± 0.29 5.06± 0.05 7.69± 0.11 2.26± 0.10 5.50± 0.25 100.00± 0.00 0.73± 0.01 0.87± 0.00 275.73± 0.42 I.2 Protein Hydrophilicity (GRAVY) Tab. 11 repeats the protein comparison with the secondary objective replaced by hydrophilicity, GRAVY <−0.70<-0.70, using the Kyte–Doolittle hydropathy scale (Kyte and Doolittle, 1982) under the same two gate modes. Two features of this task are worth isolating. First, TDS reaches 0.00%0.00\% joint success in both gate modes, despite clearing the binding threshold on approximately 52–54% of its designs. No output satisfies the hydrophilicity requirement. Second, the wall-clock axis runs the other way here. DiMOS costs about 8080 ms per design in both gate modes, below every published baseline budget, so the budget-matched partners are Best-of-N at K=8K=8 and SVDD at M=4M=4 rather than the larger settings used elsewhere. At that budget DiMOS reaches 27.39% against 18.73% and 19.12%, and it also exceeds the strongest baseline at any budget we measured. Both GRAVY configurations use M=2M=2 and K=2K=2. Table 11: Protein binder design, ΔΔG>0 G>0 and GRAVY <−0.70<-0.70. Same gate modes and protocol as Table 2. Best-of-N and SVDD are shown at the budget closest to DiMOS’s wall clock, which in the hard-gate block is below their published budgets because DiMOS runs at 7979 ms per design there. Time is milliseconds per design. Method Joint SR (%) ↑ ΔΔG G ↑ GRAVY ↓ Uniq. (%) scRMSD (%) pLDDT (%) Time (ms) Median Top-10 Median Top-10 Hard-Gate Mode Pretrained 6.84± 0.21 -0.45± 0.00 0.28± 0.02 -0.57± 0.00 -0.77± 0.00 99.67± 0.05 93.55± 0.11 87.98± 0.11 16.90± 0.08 Best-of-N 18.73± 0.16 -0.28± 0.00 0.44± 0.02 -0.75± 0.00 -0.86± 0.00 98.31± 0.09 93.55± 0.05 87.96± 0.14 81.64± 0.09 TDS 0.00± 0.00 0.18± 0.03 0.36± 0.04 0.43± 0.01 0.24± 0.01 11.11± 0.27 88.76± 0.35 84.38± 0.00 121.42± 0.75 SMC 16.93± 1.24 -0.28± 0.01 0.40± 0.06 -0.51± 0.00 -0.75± 0.01 22.59± 0.77 94.92± 0.63 86.65± 0.37 80.37± 0.06 FK-Steering 12.89± 0.14 -0.28± 0.00 0.51± 0.01 -0.64± 0.00 -0.79± 0.01 99.18± 0.11 93.06± 0.16 87.48± 0.11 192.31± 0.03 SVDD 19.12± 0.27 -0.61± 0.03 0.33± 0.01 -0.95± 0.00 -1.00± 0.01 99.09± 0.09 92.58± 0.09 88.00± 0.27 85.79± 0.02 DiMOS (Ours) 27.39± 0.12 -0.71± 0.02 0.40± 0.02 -1.08± 0.00 -1.05± 0.00 100.00± 0.00 91.86± 0.23 84.85± 0.13 79.98± 0.19 Soft-Gate Mode Pretrained 6.84± 0.21 -0.45± 0.00 0.28± 0.02 -0.57± 0.00 -0.77± 0.00 99.67± 0.05 93.55± 0.11 87.98± 0.11 16.73± 0.10 Best-of-N 21.83± 0.27 -0.28± 0.00 0.49± 0.01 -0.76± 0.00 -0.86± 0.00 98.63± 0.09 91.60± 0.11 88.19± 0.16 81.48± 0.08 TDS 0.00± 0.00 0.12± 0.01 0.39± 0.05 0.42± 0.01 0.22± 0.00 10.37± 0.21 89.97± 0.83 85.42± 0.00 117.91± 0.01 SMC 10.13± 0.35 -0.43± 0.02 0.27± 0.04 -0.60± 0.01 -0.81± 0.00 23.61± 0.91 93.55± 0.23 87.48± 0.52 78.98± 0.17 FK-Steering 13.41± 0.09 -0.41± 0.00 0.35± 0.02 -0.71± 0.01 -0.86± 0.00 99.26± 0.03 92.25± 0.11 88.06± 0.08 192.13± 0.08 SVDD 36.22± 0.26 -0.15± 0.02 0.54± 0.01 -0.86± 0.00 -0.96± 0.00 97.55± 0.08 90.47± 0.03 88.89± 0.03 85.53± 0.03 DiMOS (Ours) 46.94± 0.05 -0.03± 0.01 0.84± 0.03 -0.98± 0.00 -1.02± 0.00 99.74± 0.05 91.04± 0.13 85.50± 0.06 80.35± 0.47 I.3 RNA Design We additionally evaluate RNA design (Lorenz et al., 2011; Li et al., 2025) on translation paired with accessibility or folding stability. The two gate modes differ in the translation term: it uses a hard gate or a soft gate, while the secondary objective retains its soft gate. Both modes use the same success criteria. Translation and Accessibility. Tab. 12 is the RNA translation ++ accessibility result, moved here for space. Both objectives use separate reward models for guidance and final evaluation: guidance uses a ConvGRU MRL reward model and a constrained-MFE opening proxy, while the table reports Optimus-5′ MRL and the exact partition-function opening energy. The gap between the two reveals discrepancies between proxy and exact structural rewards. In hard-gate mode, reward-guided baselines pass the opening proxy criterion on 14–77% of designs but the exact criterion on only 1.6–4.7%, while the corresponding rates for DiMOS are 99.7% and 85.2%. For DiMOS, replacing the translation soft gate with a hard gate raises joint SR from 78.39% to 84.11%. The held-out translation pass rate remains high (99.74% versus 98.96%), while the exact accessibility pass rate rises from 78.65% to 85.16%. This is consistent with directing more reward feedback toward the remaining structural requirement. Table 12: RNA 5′UTR design with hard and soft gates. Both blocks are the same translation ++ accessibility task with the same success criterion, and differ only in whether the translation term uses a hard gate (binary penalty) or a soft gate (smooth reward). N=128N=128 designs per seed. Both objectives are graded by held-out oracles: guidance sees a ConvGRU MRL predictor and a constrained-MFE opening proxy, while the table reports Optimus-5′ MRL and the exact partition-function opening energy −RTlogpunpaired-RT p_unpaired, a different physical quantity rather than a finer estimate of the proxy. Each baseline is shown at its measured time-closest budget rung; none runs below our budget. Mean ± s.d. over three seeds; time is milliseconds per design. Method Joint SR (%) ↑ Opening ΔG G ↓ ConvGRU-MRL ↑ Uniq. (%) Optimus-MRL ↑ Time (ms) Median Top-10 Median Top-10 Hard-Gate Mode Pretrained 0.52± 0.45 4.07± 0.17 2.01± 0.40 0.34± 0.08 0.81± 0.03 100.00± 0.00 -0.11± 0.11 10.69± 0.08 Best-of-N 1.04± 1.19 2.04± 0.08 0.73± 0.03 0.89± 0.01 0.88± 0.00 100.00± 0.00 0.55± 0.02 186.22± 0.32 FK-Steering 1.82± 1.80 2.25± 0.16 0.77± 0.26 0.81± 0.01 0.89± 0.02 100.00± 0.00 0.36± 0.04 725.52± 9.83 SMC 2.86± 4.30 0.93± 0.13 0.51± 0.10 0.93± 0.06 1.01± 0.10 28.39± 1.19 0.62± 0.04 202.26± 8.12 SVDD 1.82± 0.90 1.01± 0.09 0.44± 0.02 0.86± 0.01 0.97± 0.02 100.00± 0.00 0.50± 0.02 347.65± 3.13 DiMOS (Ours) 84.11± 3.69 0.13± 0.00 0.06± 0.01 0.98± 0.01 1.08± 0.01 100.00± 0.00 0.74± 0.01 178.24± 1.39 Soft-Gate Mode Pretrained 0.52± 0.45 4.07± 0.17 2.01± 0.40 0.34± 0.08 0.81± 0.03 100.00± 0.00 -0.11± 0.11 11.25± 0.88 Best-of-N 1.30± 1.63 1.43± 0.08 0.65± 0.02 0.82± 0.01 0.88± 0.01 100.00± 0.00 0.46± 0.00 189.46± 1.19 FK-Steering 0.26± 0.45 2.34± 0.18 0.73± 0.04 0.77± 0.02 0.92± 0.02 100.00± 0.00 0.35± 0.01 726.31± 7.12 SMC 0.00± 0.00 0.66± 0.25 0.52± 0.04 0.88± 0.07 0.96± 0.01 34.11± 3.16 0.40± 0.16 202.14± 4.76 SVDD 4.17± 1.19 0.91± 0.13 0.34± 0.05 1.07± 0.01 1.21± 0.01 100.00± 0.00 0.69± 0.02 293.86± 4.65 DiMOS (Ours) 78.39± 4.96 0.14± 0.01 0.08± 0.01 1.07± 0.00 1.13± 0.01 100.00± 0.00 0.81± 0.01 172.14± 1.06 Translation and Folding Stability. Tab. 13 repeats the RNA comparison with accessibility replaced by the 50-nt minimum free energy. Only the translation objective uses a held-out reward model: MRL uses a ConvGRU reward model for guidance and Optimus-5′ for evaluation, while the folding energy has no cheaper proxy and is therefore computed by the same ViennaRNA call on both sides. This task benefits from graded translation feedback: DiMOS’s soft-gate guidance achieves 54.17% joint SR and 62.50% held-out translation success, compared with 36.72% and 51.30% under hard-gate guidance. Together with the accessibility results, this shows that the effect of the gate depends on which requirements remain difficult to satisfy. Table 13: RNA 5′UTR design, translation ++ folding stability. As in Table 12, with accessibility replaced by the 50-nt minimum free energy. MRL is graded by the held-out Optimus-5′ predictor; MFE has no cheaper guidance proxy and is the one objective shared between guidance and grading. Mean ± s.d. over three seeds; time is milliseconds per design. Method Joint SR (%) ↑ MFE ↓ ConvGRU-MRL ↑ Uniq. (%) Optimus-MRL ↑ Time (ms) Median Top-10 Median Top-10 Hard-Gate Mode Pretrained 0.26± 0.45 -8.57± 0.53 -12.13± 1.69 0.34± 0.08 0.75± 0.02 100.00± 0.00 -0.11± 0.11 10.73± 0.23 Best-of-N 0.78± 1.35 -9.38± 0.23 -12.72± 0.88 0.85± 0.01 0.84± 0.01 100.00± 0.00 0.47± 0.02 78.81± 2.70 FK-Steering 2.08± 0.45 -11.39± 0.18 -14.29± 1.16 0.72± 0.03 0.84± 0.01 100.00± 0.00 0.32± 0.06 140.91± 1.77 SMC 18.75± 7.53 -14.77± 0.89 -16.10± 0.62 0.85± 0.01 0.85± 0.02 21.09± 4.13 0.46± 0.02 68.72± 0.93 SVDD 15.62± 2.07 -14.09± 0.21 -18.40± 0.74 0.83± 0.00 0.86± 0.01 100.00± 0.00 0.44± 0.03 74.88± 3.49 DiMOS (Ours) 36.72± 2.07 -18.36± 0.17 -20.33± 0.27 0.82± 0.01 0.88± 0.03 100.00± 0.00 0.46± 0.02 36.12± 4.43 Soft-Gate Mode Pretrained 0.26± 0.45 -8.57± 0.53 -12.13± 1.69 0.34± 0.08 0.75± 0.02 100.00± 0.00 -0.11± 0.11 10.92± 0.43 Best-of-N 1.30± 0.90 -11.32± 0.43 -14.43± 0.36 0.77± 0.02 0.77± 0.01 100.00± 0.00 0.41± 0.02 79.33± 1.58 FK-Steering 4.43± 2.26 -11.62± 0.33 -15.35± 0.53 0.74± 0.03 0.83± 0.02 100.00± 0.00 0.36± 0.03 140.44± 2.46 SMC 21.35± 19.09 -17.19± 0.60 -17.47± 0.90 0.77± 0.05 0.87± 0.02 26.04± 2.39 0.34± 0.14 70.06± 3.43 SVDD 19.27± 1.63 -14.21± 0.64 -18.31± 0.45 0.88± 0.01 0.91± 0.02 100.00± 0.00 0.51± 0.03 68.83± 1.27 DiMOS (Ours) 54.17± 0.45 -18.72± 0.14 -20.10± 0.42 0.92± 0.00 0.99± 0.01 100.00± 0.00 0.50± 0.02 35.71± 3.53 Appendix J Continuous-Score Rewards Sec. 4.1 builds the joint reward from per-property thresholds. The local resampling rule of Sec. 4.2 uses exp(R/β) (R/β) and applies directly to continuous property scores that continue to increase beyond the acceptance thresholds. Tabs. 14 and 15 repeat the DNA and protein tasks with the gated reward of Eq. (10) replaced by a weighted sum of the two property scores, standardized on frozen-prior samples. The task, the seven configurations and the success criteria are unchanged, and nothing was re-tuned. Table 14: DNA design guided by a continuous property score. The same Activity ++ ATAC task and the same seven configurations as Tab. 1, with the gated reward of Eq. (10) replaced by a weighted sum of the two property scores standardized on frozen-prior samples, Rλ=λzact+(1−λ)zATACR_λ=λ z_act+(1-λ)z_ATAC. No acceptance threshold enters the reward, and nothing was re-tuned. Joint SR is scored against the unchanged canonical criteria. λ weights the first objective and is part of the reward, not a sampler setting: at one λ every method optimizes the same RλR_λ. Because z is in units of the prior’s own standard deviation, λ=0.5λ=0.5 means equal weight in prior spread, not equal importance. Blocks are different objective functions and are not comparable to each other. Guardrails are reported without ranking. N=640N=640 per seed, mean ± s.d. over three seeds. Method Joint SR (%) ↑ Activity ↑ ATAC ↑ Uniq. (%) 3-mer r JASPAR r Time (ms) Median Top-10 Median Top-10 λ=0.25λ=0.25: ATAC-leaning Pretrained 0.00± 0.00 0.18± 0.01 2.41± 0.39 0.03± 0.00 0.65± 0.06 100.00± 0.00 -0.04± 0.01 0.29± 0.01 8.13± 0.01 Best-of-N 1.77± 0.39 1.51± 0.03 5.24± 0.26 0.41± 0.01 0.89± 0.03 100.00± 0.00 0.78± 0.00 0.91± 0.00 202.12± 0.10 TDS 3.70± 1.67 1.11± 0.11 5.28± 0.13 0.28± 0.05 0.84± 0.06 51.88± 1.38 0.37± 0.06 0.56± 0.09 204.19± 0.26 SMC 6.25± 4.69 1.35± 0.29 4.79± 0.79 0.29± 0.17 0.92± 0.01 51.98± 1.19 0.39± 0.09 0.67± 0.08 90.12± 0.31 FK-Steering 6.25± 0.44 1.47± 0.04 5.83± 0.22 0.38± 0.02 0.95± 0.01 100.00± 0.00 0.71± 0.01 0.89± 0.00 110.83± 0.59 SVDD 47.34± 2.67 4.12± 0.13 7.23± 0.13 0.90± 0.00 0.99± 0.00 100.00± 0.00 0.68± 0.01 0.91± 0.01 203.62± 0.50 DiMOS (Ours) 89.90± 1.19 5.75± 0.05 7.52± 0.18 0.99± 0.00 0.99± 0.00 100.00± 0.00 0.91± 0.01 0.93± 0.00 195.46± 0.74 λ=0.50λ=0.50: equal weight Pretrained 0.00± 0.00 0.18± 0.01 2.41± 0.39 0.03± 0.00 0.65± 0.06 100.00± 0.00 -0.04± 0.01 0.29± 0.01 8.34± 0.10 Best-of-N 1.77± 0.39 2.06± 0.01 5.24± 0.26 0.37± 0.01 0.89± 0.03 100.00± 0.00 0.77± 0.01 0.90± 0.00 207.86± 0.22 TDS 6.61± 1.98 1.05± 0.17 5.82± 0.78 0.24± 0.09 0.89± 0.04 53.12± 2.01 0.57± 0.19 0.66± 0.13 209.51± 0.40 SMC 7.40± 3.94 1.29± 0.38 5.56± 0.37 0.16± 0.03 0.85± 0.07 52.76± 0.70 0.55± 0.04 0.73± 0.04 89.10± 0.24 FK-Steering 7.71± 0.53 2.03± 0.05 5.88± 0.10 0.33± 0.01 0.94± 0.01 100.00± 0.00 0.70± 0.02 0.90± 0.00 110.64± 0.39 SVDD 49.84± 1.33 4.63± 0.12 7.26± 0.04 0.82± 0.00 0.98± 0.00 100.00± 0.00 0.76± 0.00 0.93± 0.00 203.46± 0.42 DiMOS (Ours) 92.03± 0.46 5.85± 0.03 7.40± 0.13 0.98± 0.00 0.99± 0.00 100.00± 0.00 0.93± 0.01 0.94± 0.00 195.50± 0.76 λ=0.75λ=0.75: Activity-leaning Pretrained 0.00± 0.00 0.18± 0.01 2.41± 0.39 0.03± 0.00 0.65± 0.06 100.00± 0.00 -0.04± 0.01 0.29± 0.01 8.28± 0.01 Best-of-N 1.77± 0.39 2.38± 0.02 5.24± 0.26 0.25± 0.00 0.89± 0.03 100.00± 0.00 0.68± 0.02 0.85± 0.01 202.70± 0.22 TDS 8.54± 4.41 2.03± 0.76 5.49± 0.14 0.18± 0.07 0.90± 0.06 53.33± 0.70 0.57± 0.21 0.65± 0.11 208.40± 0.17 SMC 5.52± 5.22 1.13± 0.28 5.42± 0.35 0.13± 0.02 0.87± 0.09 52.50± 0.34 0.39± 0.14 0.61± 0.14 88.47± 0.32 FK-Steering 6.61± 0.32 2.10± 0.09 5.94± 0.17 0.20± 0.01 0.94± 0.02 100.00± 0.00 0.62± 0.03 0.85± 0.02 110.55± 0.38 SVDD 38.70± 1.47 4.98± 0.02 7.17± 0.05 0.62± 0.03 0.98± 0.00 100.00± 0.00 0.77± 0.00 0.94± 0.00 203.41± 0.40 DiMOS (Ours) 91.77± 0.52 5.89± 0.00 7.60± 0.11 0.97± 0.00 0.99± 0.00 100.00± 0.00 0.95± 0.00 0.94± 0.00 195.48± 0.72 Table 15: Protein design guided by a continuous property score. The same ΔΔG G ++ SeqRec task and the same seven configurations as Tab. 2, with the gated reward replaced by Rλ=λzΔΔG+(1−λ)zSeqRecR_λ=λ z_ G+(1-λ)z_SeqRec on frozen-prior statistics. No acceptance threshold enters the reward, and nothing was re-tuned. Joint SR is scored against the unchanged ΔΔG>0 G>0 and SeqRec >0.55>0.55. λ weights ΔΔG G and is part of the reward, not a sampler setting. Because z is in units of the prior’s own standard deviation, λ=0.5λ=0.5 means equal weight in prior spread, not equal importance. Blocks are different objective functions and are not comparable to each other. Guardrails are reported without ranking. N=128N=128 per backbone over 12 backbones, mean ± s.d. over three seeds. Method Joint SR (%) ↑ ΔΔG G ↑ SeqRec ↑ Uniq. (%) scRMSD (%) pLDDT (%) Time (ms) Median Top-10 Median Top-10 λ=0.25λ=0.25: SeqRec-leaning Pretrained 4.04± 0.30 -0.45± 0.00 0.49± 0.03 0.44± 0.00 0.48± 0.00 99.67± 0.05 93.55± 0.11 87.98± 0.11 17.36± 0.04 Best-of-N 15.30± 0.19 0.03± 0.01 0.80± 0.02 0.50± 0.00 0.52± 0.00 99.15± 0.16 93.36± 0.16 86.46± 0.16 160.54± 0.12 TDS 3.84± 0.00 -0.47± 0.04 0.61± 0.11 0.45± 0.00 0.47± 0.00 21.70± 0.72 95.25± 0.42 89.34± 0.15 126.66± 0.03 SMC 6.60± 0.43 -0.40± 0.02 0.42± 0.01 0.45± 0.00 0.48± 0.00 23.65± 0.80 93.47± 0.62 86.98± 0.14 84.37± 0.84 FK-Steering 4.38± 0.06 -0.46± 0.00 0.57± 0.02 0.45± 0.00 0.49± 0.00 99.72± 0.08 93.29± 0.19 86.65± 0.11 193.09± 0.01 SVDD 33.59± 0.33 0.32± 0.01 0.98± 0.01 0.54± 0.00 0.57± 0.00 97.29± 0.22 95.51± 0.18 84.24± 0.05 167.18± 0.10 DiMOS (Ours) 57.60± 0.27 0.42± 0.00 1.16± 0.02 0.61± 0.00 0.66± 0.00 100.00± 0.00 92.56± 0.03 83.92± 0.05 158.24± 0.02 λ=0.50λ=0.50: equal weight Pretrained 4.04± 0.30 -0.45± 0.00 0.58± 0.02 0.44± 0.00 0.47± 0.00 99.67± 0.05 93.55± 0.11 87.98± 0.11 17.50± 0.02 Best-of-N 15.45± 0.11 0.14± 0.01 0.92± 0.01 0.48± 0.00 0.50± 0.00 99.35± 0.16 93.45± 0.06 86.02± 0.11 160.49± 0.15 TDS 4.36± 0.98 -0.46± 0.06 0.72± 0.06 0.44± 0.00 0.46± 0.00 22.24± 0.20 94.12± 0.35 88.00± 0.55 125.66± 0.16 SMC 5.95± 0.34 -0.55± 0.06 0.34± 0.03 0.44± 0.01 0.48± 0.00 23.39± 0.93 92.62± 0.85 86.48± 0.46 83.71± 0.59 FK-Steering 4.36± 0.16 -0.43± 0.00 0.63± 0.01 0.45± 0.00 0.48± 0.00 99.39± 0.03 92.90± 0.16 87.22± 0.08 193.28± 0.04 SVDD 31.58± 0.11 0.46± 0.00 1.15± 0.00 0.53± 0.00 0.55± 0.00 97.46± 0.05 94.23± 0.17 84.57± 0.05 166.98± 0.10 DiMOS (Ours) 46.05± 0.41 0.58± 0.01 1.28± 0.00 0.57± 0.00 0.61± 0.00 99.93± 0.05 93.29± 0.05 84.74± 0.08 162.25± 5.94 λ=0.75λ=0.75: ΔΔG G-leaning Pretrained 4.04± 0.30 -0.45± 0.00 0.64± 0.02 0.44± 0.00 0.47± 0.00 99.67± 0.05 93.55± 0.11 87.98± 0.11 17.28± 0.06 Best-of-N 13.65± 0.08 0.24± 0.00 1.01± 0.00 0.46± 0.00 0.49± 0.00 99.13± 0.13 93.88± 0.18 85.83± 0.16 159.62± 0.13 TDS 5.88± 0.30 -0.34± 0.01 0.54± 0.03 0.44± 0.00 0.47± 0.00 23.81± 0.86 91.51± 0.39 88.85± 0.06 119.42± 0.17 SMC 2.99± 0.51 -0.56± 0.01 0.44± 0.04 0.44± 0.00 0.47± 0.01 24.76± 1.25 93.90± 0.13 87.65± 0.34 80.04± 0.30 FK-Steering 5.92± 0.00 -0.36± 0.00 0.66± 0.01 0.45± 0.00 0.47± 0.00 99.22± 0.11 93.29± 0.23 87.54± 0.20 192.22± 0.01 SVDD 16.91± 0.16 0.57± 0.01 1.31± 0.01 0.49± 0.00 0.52± 0.00 98.03± 0.20 95.59± 0.21 85.05± 0.17 166.82± 0.10 DiMOS (Ours) 19.51± 0.27 0.76± 0.01 1.40± 0.01 0.50± 0.00 0.54± 0.00 100.00± 0.00 93.90± 0.12 85.26± 0.03 158.06± 0.07