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D-DOIT: Training-free Adaptation of Discrete Diffusion via Doob's h-Transform
Jieke Wu, Qijie Zhu, Weimin Wu, Zeqi Ye, Minshuo Chen, Han Liu
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Abstract
Abstract:We propose D-DOIT (Discrete Doob-Oriented Inference-time Transformation), a training-free and efficient adaptation method for discrete diffusion models with generic rewards. D-DOIT formulates adaptation as sampling from a reward-tilted target distribution and realizes this transport through Doob's h-transform of the discrete diffusion reverse kernel, using only reward values rather than reward gradients. Unlike continuous diffusion, masked discrete diffusion samples categorical token-reveal transitions rather than continuous state updates. D-DOIT derives the corresponding discrete Doob's h-transform, which guides sampling by reweighting reverse transition probabilities instead of adding a drift correction. To make this transformation practical, D-DOIT avoids expensive future rollouts. At each guided step, D-DOIT samples candidate next states, uses the model prediction head to complete each candidate into a clean sequence, evaluates each completion with the reward oracle, and resamples the next state with probabilities proportional to the rewards. An optional late-stage best-of-K refinement further improves sample quality by branching trajectories only near the end of denoising, avoiding the $K$-fold cost over the full trajectory. Empirically, across regulatory DNA design and protein inverse folding benchmarks, D-DOIT outperforms training-free guidance baselines. It improves enhancer activity and cell-type specificity while preserving sequence naturalness, and achieves the highest success rate in protein inverse folding.
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- Source: https://arxiv.org/abs/2610.04938v1
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D-DOIT: Training-free Adaptation of Discrete Diffusion via Doob’s h-Transform Jieke Wu‡^\>\; , Qijie Zhu*§, Weimin Wu*†, Zeqi Ye♮, Minshuo Chen♮, Han Liu†§ † \;Center for Foundation Models and Generative AI, Northwestern University Department of Computer Science, Northwestern University ‡ \;Department of Computer Science, King Abdullah University of Science and Technology § @sectionsign\;Department of Statistics and Data Science, Northwestern University ♮ \;Industrial Engineering & Management Sciences, Northwestern University jieke.wu@kaust.edu.sa, qijiezhu2029,wwm,zeqiye2029@u.northwestern.edu minshuo.chen@northwestern.edu, hanliu@northwestern.edu †thanks: Equal contribution. Abstract We propose D-DOIT (Discrete Doob-Oriented Inference-time Transformation), a training-free and efficient adaptation method for discrete diffusion models with generic rewards. D-DOIT formulates adaptation as sampling from a reward-tilted target distribution and realizes this transport through Doob’s h-transform of the discrete diffusion reverse kernel, using only reward values rather than reward gradients. Unlike continuous diffusion, masked discrete diffusion samples categorical token-reveal transitions rather than continuous state updates. D-DOIT derives the corresponding discrete Doob’s h-transform, which guides sampling by reweighting reverse transition probabilities instead of adding a drift correction. To make this transformation practical, D-DOIT avoids expensive future rollouts. At each guided step, D-DOIT samples candidate next states, uses the model prediction head to complete each candidate into a clean sequence, evaluates each completion with the reward oracle, and resamples the next state with probabilities proportional to the rewards. An optional late-stage best-of-K refinement further improves sample quality by branching trajectories only near the end of denoising, avoiding the K-fold cost over the full trajectory. Empirically, across regulatory DNA design and protein inverse folding benchmarks, D-DOIT outperforms training-free guidance baselines. It improves enhancer activity and cell-type specificity while preserving sequence naturalness, and achieves the highest success rate in protein inverse folding. Project Page: https://github.com/WeiminWu2000/D-DOIT. 1 Introduction We propose D-DOIT (Discrete Doob-Oriented Inference-time Transformation), a training-free sampler for reward-guided masked discrete diffusion. Discrete diffusion has become a strong generative modeling paradigm for finite-vocabulary data [1, 2]. Masked discrete diffusion models are especially attractive for biological sequences because they predict clean tokens from partially masked states and provide tractable categorical reverse kernels [3, 4, 5]. These models have recently been used for DNA and protein generation [6, 7, 8], where the scientific objective is rarely just to sample from the pre-trained distribution. Instead, a practitioner usually wants sequences with high activity, specificity, or other task-level scores for sequence design. Reward adaptation creates a tension between performance, generality, and cost. Training-based approaches can improve reward optimization, but they update the backbone and often require reward gradients, extra training, and hyperparameter tuning [8, 9]. Training-free methods keep the backbone fixed, but many either assume differentiable rewards or rely on particle and search procedures whose cost grows with trajectory history or search width [10, 11, 12, 13, 14, 15, 16, 17]. This motivates the question studied here: can a frozen masked diffusion model be adapted to generic black-box rewards by modifying only its inference-time sampling law? Our starting point is to view adaptation as a distributional transport problem. Let pθp_θ be the clean-sequence distribution induced by the frozen masked diffusion reverse chain, and let R:→ℝR:X be a sequence-level reward. Rather than optimizing reward in isolation, we target the reward-tilted distribution πβ()∝pθ()exp(R()/β) _β(x) p_θ(x) (R( )/β), which keeps the pre-trained model as a sequence prior while favoring high-reward outputs. Doob’s h-transform gives the exact local transformation needed to sample from this tilted law. The central object is the downstream value h(¯tl,tl)h( x_t_l,t_l), the expected terminal reward weight under the frozen reverse chain conditioned on the current partially denoised state. The discrete setting changes how the transform must be implemented. In continuous diffusion, Doob guidance can be expressed as an additive score or drift correction [18, 19, 20, 21]. In masked discrete diffusion, the state space is categorical, generation proceeds through irreversible token-reveal transitions, and there is no Euclidean score field. D-DOIT therefore applies Doob’s h-transform at the kernel level: each frozen reverse transition is multiplicatively reweighted by the downstream h-value of the candidate next state. This yields a discrete analogue of the continuous Doob correction while preserving the frozen parameters and allowing black-box reward queries. The exact transform still requires the future reward-to-go for every candidate reverse state, which is intractable by repeated rollouts. D-DOIT makes the transform executable with two approximations. First, it estimates candidate h-values by using the masked diffusion prediction head to complete each candidate into a clean sequence, then evaluating the reward oracle on that completion. Second, it draws a finite set of candidate reverse states and selects among them with Doob-weighted probabilities. An optional best-of-K refinement branches only after a chosen split update, which hedges irreversible token reveals without multiplying the full denoising trajectory cost. The same sampler gives two operating points: D-DOIT-LBOK-S for a lower-budget regime and D-DOIT-Q for high-budget quality evidence. We evaluate D-DOIT on regulatory DNA sequence design and protein inverse folding. On the DRAKES HepG2 enhancer benchmark, the near-TDS-speed D-DOIT-LBOK-S matches TDS on predicted activity while substantially improving accessibility, 3-mer correlation, and motif correlation. The high-budget D-DOIT-Q setting reaches 6.196.19 predicted activity and exceeds training-based DRAKES on Pred-Activity, 3-mer Corr, and JASPAR Corr, while remaining slightly lower on ATAC-Acc. At a similar wall-clock budget, D-DOIT-Q also improves activity, accessibility, and 3-mer correlation over soft value-based decoding (SVDD-PM) and wall-clock-matched endpoint search. Endpoint search achieves higher JASPAR correlation. On the Ctrl-DNA cell-type-specific enhancer benchmark, D-DOIT is a clear training-free method that moves the specificity reward. D-DOIT-Q raises ΔR R to 0.1190.119, with a trade-off in motif preservation, and D-DOIT-LBOK-S provides a lower-budget positive-reward point. At a matched wall-clock budget, however, endpoint search captures much of this gain, so the margin is smaller than on HepG2. Guidance with alternative predictors and transfer to held-out evaluators retain the gains over unguided sampling. On protein inverse folding, the same frozen backbone reaches 68.1%68.1\% joint stability–structure success. These results support D-DOIT as a training-free mechanism for reward-guided sequence generation, not as a matched-cost replacement for every high-budget or training-based method. In summary, we make three contributions: • Discrete Doob h-transform for masked diffusion. We formulate reward adaptation as sampling from a reward-tilted target distribution and derive the corresponding Doob h-transform of the masked discrete diffusion reverse kernel. The resulting guidance reweights categorical token-reveal transitions, in contrast to the additive drift or score correction used in continuous diffusion. The exact reference transform is kept separate from the executable approximations built on it. • Executable finite-candidate sampler. We make the exact transform practical with a prediction-head h-estimator, finite-candidate Doob-weighted selection, and an optional best-of-K refinement for irreversible reveal decisions. The method uses only reward values and does not update the diffusion backbone. We characterize how tempering, sharpening, and terminal selection deviate from the exact target. • Biological validation and trade-offs. We test the same frozen-backbone sampler on HepG2 enhancer design, cell-type-specific enhancer design, and protein inverse folding. D-DOIT improves over training-free baselines across the reported reward-guided design tasks, while preserving the relevant trade-offs in naturalness, motif structure, and inference budget. Organization. Section 2 reviews the masked discrete diffusion backbone, defines the reward-tilted target, and states the exact discrete Doob h-transform. Section 3 develops the practical D-DOIT sampler, including the prediction-head h-estimator, finite-candidate weighted update, and optional best-of-K refinement. Section 4 reports experiments on regulatory DNA design, cell-type-specific enhancer design, and protein inverse folding. The appendix provides related work (Appendix A), the proof of the discrete Doob h-transform (Appendix B), the standard masked diffusion sampler (Appendix E), and full experimental details (Appendices J, K and M). Notation. We write =A,G,T,CV=\A,G,T,C\ for the clean token alphabet, d=||d=|V|, and =NX=V^N for length-N DNA sequences. Let m be the mask token, ~=∪ V=V∪\ m\, and ~=~N X= V^N. We use tlX_t_l for the forward process and ¯tl X_t_l for the learned reverse chain, which runs from the all-Mask state ¯tL x_t_L to a clean sequence ¯0 x_0. We use q for the forward corruption kernel and analytic posterior, Pθ,tlP_θ,t_l for the learned reverse kernel, pθ,tlp_θ,t_l for the reverse-chain marginal, and R:→ℝR:X with ϕ(0)=exp(R(0)/β)φ(x_0)= (R( _0)/β) for the black-box reward and reward weight. For the sampler, M denotes the local candidate budget, nhn_h the number of completions per candidate, τ and γ control proposal and completion tempering, ω sharpens value-based selection, and K denotes the best-of-K branching width. 2 Preliminaries We first define the frozen masked diffusion reverse chain that serves as the base law for D-DOIT. We then state the reward-tilted target distribution and the discrete Doob h-transform that transports this base law to the target without changing the model parameters [22, 23]. We keep the exact reference transform separate from the executable approximation built on it, which Section 3 develops. 2.1 Masked Discrete Diffusion Models We specialize to masked discrete diffusion. Let V be the clean token alphabet, d=||d=|V|, and m a special mask token. Define ~:=∪ V:=V∪\ m\, with sequence spaces =NX=V^N and ~=~N X= V^N. The data distribution is supported on X, while the forward process evolves on ~ X by progressively replacing clean tokens with m. On the grid tl=lT/Lt_l=lT/L, let bl∈[0,1]b_l∈[0,1] be the masking probability on [tl−1,tl][t_l-1,t_l]. The single-token forward kernel is qtl−1,tl(y∣x)=(1−bl)y=x+bly=,x,y∈~, q_t_l-1,t_l(y x)=(1-b_l)1\y=x\+b_l1\y= m\, x,y∈ V, so with probability 1−bl1-b_l the token stays fixed and otherwise it jumps to the mask state; once masked it stays masked. With αl=∏r=1l(1−br) _l= _r=1^l(1-b_r), a clean token x0∈x_0 satisfies q0,tl(y∣x0)=ℙ(Xtl=y∣X0=x0)=αly=x0+(1−αl)y=. q_0,t_l(y x_0)=P(X_t_l=y X_0=x_0)= _l1\y=x_0\+(1- _l)1\y= m\. We choose αL≈0 _L≈ 0 for a sufficiently large T, so that q0,Tq_0,T is concentrated near the mask state. The forward masking process is applied independently across coordinates. The conditional one-step time reversal also has a closed form, and Appendix F records the forward transition matrices and the analytic reverse kernel that the learned sampler approximates. Since 0x_0 is unknown during generation, masked diffusion models approximate the clean-token posterior with a trained neural network. For each coordinate j, let μθj(⋅∣tl,tl)∈Δ() _θ^j(· _t_l,t_l)∈ (V) denote the neural network’s approximation to ℙ(X0j=⋅∣tl=tl)P(X_0^j=· _t_l=x_t_l). Plugging this approximation into the analytic reverse posterior gives the learned one-step reverse transition from tlt_l to tl−1t_l-1: Pθ,tl(xtl−1j=y∣tl)=(1−αl−1)/(1−αl),xtlj=,y=,(αl−1−αl)/(1−αl)μθj(y∣tl,tl),xtlj=,y∈,y=xtlj,xtlj≠. P_θ,t_l(x_t_l-1^j=y _t_l)= cases(1- _l-1)/(1- _l),&x_t_l^j= m,\ y= m,\\ ( _l-1- _l)/(1- _l)\, _θ^j(y _t_l,t_l),&x_t_l^j= m,\ y ,\\ 1\y=x_t_l^j\,&x_t_l^j≠ m. cases (2.1) Finally, the learned reverse model factorizes over coordinates, Pθ,tl(tl−1∣tl) P_θ,t_l(x_t_l-1 _t_l) =∏j=1NPθ,tl(xtl−1j∣tl). = _j=1^NP_θ,t_l(x_t_l-1^j _t_l). Starting from an initial masked state ¯tL X_t_L, we iteratively sample ¯tl−1∼Pθ,tl(⋅∣¯tl) X_t_l-1 P_θ,t_l(· X_t_l) from l=Ll=L to l=1l=1 and obtain the final clean sample ¯0 X_0. Let pθ,tlp_θ,t_l denote the marginal law of ¯tl X_t_l under this reverse chain, and write pθ=pθ,0p_θ=p_θ,0 for the clean-time law. This learned reverse chain is the frozen base sampler. D-DOIT leaves its parameters fixed and modifies only its reverse transition probabilities at inference time. We next define the target these modified transitions should realize. 2.2 Reward-tilted Target and Doob’s h-transform We now formalize the inference-time adaptation objective. Given a reward function R:→ℝR:X and a temperature parameter β>0β>0, define the positive reward weight ϕ(0)=exp(R(0)/β). φ(x_0)= (R( _0)/β). (2.2) The desired clean-time target is the reward-tilted version of the frozen model law, πβ(0) _β(x_0) =pθ(0)ϕ(0)Zβ,Zβ=0∼pθ[ϕ(0)]. = p_θ(x_0)φ(x_0)Z_β, Z_β=E_X_0 p_θ [φ(X_0) ]. (2.3) Thus adaptation is a distributional problem: change the terminal law from pθp_θ to πβ _β while preserving the frozen reverse chain as the base proposal. The discrete Doob h-function is the object that makes this transport local in time and gives a kernel-level correction. Definition 2.1 (Discrete Doob h-function). For the frozen reverse chain defined above, the discrete Doob h-function is the downstream reward weight conditioned on a partially denoised state: h(¯tl,tl)=pθ[exp(R(0)/β)∣¯tl=¯tl],l=0,…,L. h( x_t_l,t_l)=E_p_θ [ (R( _0)/β) X_t_l= x_t_l ], l=0,…,L. (2.4) At clean time, this reduces to h(0,0)=ϕ(0)h(x_0,0)=φ(x_0). At an intermediate time, h(¯tl,tl)h( x_t_l,t_l) is the reward-to-go under the frozen reverse chain. It therefore measures how promising the current partial sequence is for the terminal reward objective at clean time under the frozen process. The next theorem states the exact kernel transform induced by this value function. It is the discrete analogue of the continuous Doob correction, but it acts by reweighting transition probabilities rather than by adding a vector field to a continuous score. Theorem 2.1 (Discrete Doob’s h-transform). In the fully masked terminal setting, ¯tL X_t_L is deterministic, so h(¯tL,tL)=Zβh( X_t_L,t_L)=Z_β and the terminal distribution is unchanged. Let h be defined as in Definition 2.1. Then Doob’s h-transform induces the reverse kernel Pθ,tlh(¯tl−1∣¯tl)=Pθ,tl(¯tl−1∣¯tl)h(¯tl−1,tl−1)h(¯tl,tl),l=1,…,L. P_θ,t_l^h( x_t_l-1 x_t_l)=P_θ,t_l( x_t_l-1 x_t_l) h( x_t_l-1,t_l-1)h( x_t_l,t_l), l=1,…,L. (2.5) The resulting Doob-transformed reverse chain has marginal distribution pθ,tlh(¯tl)=pθ,tl(¯tl)h(¯tl,tl)Zβ,l=0,…,L, p_θ,t_l^h( x_t_l)= p_θ,t_l( x_t_l)h( x_t_l,t_l)Z_β, l=0,…,L, In particular, since h(0,0)=exp(R(0)/β), h(x_0,0)= (R( _0)/β), the clean-time marginal of the Doob-transformed reverse chain is exactly the reward-tilted distribution pθ,0h(¯0)=πβ(¯0)=pθ,0(¯0)exp(R(¯0)/β)Zβ. p_θ,0^h( x_0)= _β( x_0)= p_θ,0( x_0) (R( _0)/β)Z_β. The proof is deferred to Appendix B. The theorem identifies the exact population object that D-DOIT approximates. If the downstream value ratio h(¯tl−1,tl−1)/h(¯tl,tl)h( x_t_l-1,t_l-1)/h( x_t_l,t_l) were available at every reverse step, then local reweighting of the frozen transition kernel would sample exactly from πβ _β at clean time. This guarantee is a statement about the true rollout value and the exact normalized kernel, not the prediction-head implementation. In Section 3, the head proxy, finite candidate set, proposal and completion temperatures, weight sharpening, and terminal selection each have a distinct role. Finite-candidate convergence alone cannot remove a systematic value or proposal mismatch. We keep the original P-referenced πβ _β as the theoretical benchmark; Appendix B separately states the exact transform for a temperature-modified reference chain. The practical method in Section 3 estimates this ratio without retraining the masked diffusion model. The continuous analogue appears as an additive score correction ∇logpt+∇loght∇ p_t+∇ h_t. The categorical and irreversible reverse dynamics of masked diffusion have no Euclidean score field, and instead require the multiplicative kernel reweighting in (2.5); Appendix C records the continuous derivation. 3 D-DOIT: A Discrete Sampler Based on Doob’s h-transform In this section, we introduce D-DOIT, a training-free sampler that guides a frozen masked discrete diffusion model toward the reward-tilted target in (2.3) using black-box rewards R:→ℝR:X defined on clean sequences. The exact transform in Theorem 2.1 says what the sampler should do: reweight each frozen reverse kernel by the downstream value ratio of the discrete Doob h-function. The challenge is computational, since evaluating this value exactly would require future denoising rollouts from every candidate next state. D-DOIT therefore follows an exact-object-to-estimator design: it estimates the downstream value with the frozen prediction head, approximates the transformed kernel with finite candidates, and optionally branches late to hedge irreversible token reveals. Figure 1 summarizes the construction. Figure 1: Overview of D-DOIT. Left: Standard masked diffusion unmasks positions independently at each reverse step (2.1); it is cheap but reward-unaware. Right: One guided step approximately tilts the reverse kernel: it draws M=3M=3 candidates from a tempered proposal (3.6), completes and scores them with the prediction head and reward to obtain proxy h-values, and selects the next state with sharpened h-weights (3.7). Bottom: Optional best-of-K refinement duplicates each trajectory into K=4K=4 copies after the split and keeps the highest-reward suffix (3.9), sharing prefix computation. Revealed tokens stay fixed; the practical sampler does not inherit the exact-target guarantee. 3.1 D-DOIT Algorithm The transformed kernel in (2.5) depends on the downstream value h(¯tl−1,tl−1)h( y_t_l-1,t_l-1) for each candidate reverse state ¯tl−1∈~ y_t_l-1∈ X reachable from the current state ¯tl x_t_l. This subsection turns that exact identity into an executable sampler. The first component estimates each candidate’s downstream value with a prediction-head completion. The second component uses these values to form a finite-candidate approximation to the transformed kernel. The final component adds an optional best-of-K refinement for the irreversible reveal structure of masked diffusion. Prediction-head h-estimator. The exact value h(¯tl−1,tl−1)h( y_t_l-1,t_l-1) averages the reward weight over all complete denoising continuations from the candidate next state ¯tl−1 y_t_l-1. Naive Monte Carlo would roll out the frozen reverse chain many times from every candidate. D-DOIT avoids these nested rollouts by evaluating candidates with the masked discrete diffusion prediction head at ¯tl−1 y_t_l-1. Recall from Section 2.1 that μθj(⋅∣tl,tl)∈Δ() _θ^j(· _t_l,t_l)∈ (V) denotes the neural network’s approximation to the clean-token posterior at coordinate j. The head provides a direct clean-sequence completion distribution by fixing revealed tokens and independently sampling still-masked positions from the coordinate-wise clean-token posteriors μθj(⋅∣¯tl−1,tl−1) _θ^j(· y_t_l-1,t_l-1); we write pθheadp^head_θ for this direct completion law, which is not generally a rollout conditional: pθhead(0∣¯tl−1)=∏j:y¯tl−1j≠δy¯tl−1j(x0j)∏j:y¯tl−1j=μθj(x0j∣¯tl−1,tl−1). p^head_θ(x_0 y_t_l-1)= _ j:\, y_t_l-1^j≠ m _ y_t_l-1^j(x_0^j) _ j:\, y_t_l-1^j= m _θ^j(x_0^j y_t_l-1,t_l-1). (3.1) Definition 3.1 (Prediction-head h-estimator). For every reachable candidate reverse state ¯tl−1∈~ y_t_l-1∈ X, D-DOIT estimates the downstream value by the one-step clean-completion expectation under the factorized prediction-head distribution for each candidate state: h¯θ(¯tl−1,tl−1) h_θ( y_t_l-1,t_l-1) =0∼pθhead(0∣¯tl−1)[ϕ(0)]. =E_x_0 p^head_θ(x_0 y_t_l-1)\! [φ(x_0) ]. (3.2) The estimator (3.2) replaces the rollout value h(¯tl−1,tl−1)h( y_t_l-1,t_l-1) from Definition 2.1 with a single-step completion value. This is the main approximation that makes the transform practical; Appendix D discusses its bias relative to full future rollouts. For each candidate, we estimate (3.2) with one tempered completion ^0∼pθ,γhead(⋅∣¯tl−1) x_0 p^head_θ,γ(· y_t_l-1): h^(¯tl−1,tl−1) h( y_t_l-1,t_l-1) =exp(R(^0)/β),^0∼pθ,γhead(⋅∣¯tl−1), = (R( _0)/β), x_0 p^head_θ,γ(· y_t_l-1), (3.3) μθ,γj(v∣¯tl−1,tl−1) _θ,γ^j(v y_t_l-1,t_l-1) ∝μθj(v∣¯tl−1,tl−1)1/γ. _θ^j(v y_t_l-1,t_l-1)^1/γ. (3.4) The completion temperature γ tempers the completion of a candidate ¯tl−1 y_t_l-1 into a clean sequence. Smaller γ sharpens the completion toward the model mode, while larger γ smooths noisy predictions. Because the reward is queried only on the sampled prediction-head completion ^0 x_0, R(⋅)R(·) need not be differentiable, and no gradients are backpropagated through the reward oracle. Using nhn_h independent completions per candidate averages h^nh(¯tl−1,tl−1) h_n_h( y_t_l-1,t_l-1) =1nh∑i=1nhexp(R(^0(i))/β),^0(i)∼pθ,γhead(⋅∣¯tl−1), = 1n_h _i=1^n_h (R( _0^(i))/β), x_0^(i) p^head_θ,γ(· y_t_l-1), (3.5) which reduces Monte Carlo variance for this head expectation without removing its discrepancy from the rollout value. The reported main configurations use nh=1n_h=1. Finite-candidate discrete h-transform. At reverse step l, D-DOIT first draws M candidate next states ¯tl−1(m)m=1M\ y_t_l-1^(m)\_m=1^M from a τ-tempered masked discrete diffusion proposal: μθ,τj(v∣¯tl,tl)∝μθj(v∣¯tl,tl)1/τ,v∈. _θ,τ^j(v x_t_l,t_l) _θ^j(v x_t_l,t_l)^1/τ, v . (3.6) Both temperatures act on the same coordinate-wise prediction head μθj _θ^j, but they play different roles: τ controls proposal diversity, while γ controls the clean completion used for reward scoring. We draw each candidate ¯tl−1(m) y_t_l-1^(m) from the masked discrete diffusion reverse kernel (2.1) with pθ,τp_θ,τ in place of pθp_θ, keeping the schedule factor (αl−1−αl)/(1−αl)( _l-1- _l)/(1- _l) and the copy-through rule. All M proposals share one forward pass, and each is completed once via (3.3). The next state ¯tl−1=¯tl−1(m∗) x_t_l-1= y_t_l-1^(m^*) is drawn by sampling the index m∗m^* over the M candidates, each assigned probability w(m)=h^(¯tl−1(m),tl−1)ω∑r=1Mh^(¯tl−1(r),tl−1)ω. w^(m)= h( y_t_l-1^(m),t_l-1)^ω _r=1^M h( y_t_l-1^(r),t_l-1)^ω. (3.7) With τ=1τ=1, (3.7) is a finite-candidate self-normalized approximation to the h-reweighted transition induced by (2.5), with ω controlling how sharply the sampler follows the estimated downstream value. For one completion, h^ω=exp(ωR(^0)/β) h^ω= (ω R( _0)/β), so ω is not a separate approximation. Every (ω,β)(ω,β) pair is equivalent to ω=1ω=1 with effective inverse temperature η=ω/βη=ω/β, the parameter used in Appendix B. With τ>1τ>1, the proposal is deliberately smoothed, so the weighted selection rule becomes a biased proposal approximation that we use to maintain candidate diversity. Retaining the original P-referenced target under a smoothed proposal would instead require the importance weight w~(m)∝Pθ,tl(¯tl−1(m)∣¯tl)Qlτ(¯tl−1(m)∣¯tl)h(¯tl−1(m),tl−1), w^(m) P_θ,t_l( y_t_l-1^(m) x_t_l)Q^τ_l( y_t_l-1^(m) x_t_l)h( y_t_l-1^(m),t_l-1), (3.8) where QlτQ^τ_l is the proposal kernel, assuming its support covers the target transition. Our operating points omit this ratio, so it is a reference correction, not the executed algorithm. Even with it, a biased proxy, sharpening, and terminal best-of-K selection would remain. A guided step costs 1+M1+M model-side NFE: one shared proposal pass and M completion passes. It also costs M reward evaluations; Appendix G gives the full accounting and cost conventions. Best-of-K refinement. Revealed tokens cannot be revised, so D-DOIT optionally duplicates each active trajectory into K copies after split update ssplits_split, counted in reverse updates. The copies continue independently, and the method keeps the highest-reward clean sequence within each group. For each original trajectory b, the selected output is 0(b)=0(b,kb∗),kb∗=argmaxk∈1,…,KR(0(b,k)). _0^(b)=x_0^(b,k_b^*), k_b^*= argmax_k∈\1,…,K\R(x_0^(b,k)). (3.9) The K×K× multiplier is paid only after the split, hedging early decisions without always multiplying the full trajectory cost. This terminal argmax changes the output law and cannot repair earlier reveals. Before the split a trajectory is shared; after it the K copies update independently, and with K=1K=1 or ssplit=Ls_split=L the method reduces to D-DOIT. Appendices H and H in the appendix give the sampler and its per-step guided update. 3.2 Two Configurations: D-DOIT-LBOK-S and D-DOIT-Q We report two operating points of the same sampler that differ only in budget allocation. D-DOIT-LBOK-S uses a small per-step candidate budget plus late best-of-K refinement for the practical lower-budget regime. D-DOIT-Q allocates more inference budget to candidate evaluation and/or split-time refinement to test the quality frontier of the frozen backbone. Table 5 lists the benchmark-specific values, and Section 4 evaluates both configurations against training-free and training-based baselines. 4 Experiments In this section, we include the following results: (1) regulatory DNA enhancer design (Section 4.1); (2) cell-type-specific enhancer design (Section 4.2); (3) protein inverse folding (Section 4.3); (4) contrast with the closest reward-guided sampling baselines (Section 4.4); (5) analysis of the estimation accuracy of ^0 x_0 (Section 4.5); (6) analysis of the impact of different reward functions (Section 4.6). Table 1: Comparison of reward-guided sampling methods for discrete diffusion. We contrast D-DOIT with our guidance baselines (Section 4); CG = classifier guidance, TDS = twisted diffusion sampler, SMC = sequential Monte Carlo, SVDD = soft value-based decoding. Property CG TDS/SMC SVDD D-DOIT (Ours) Reward intervention gradient (backward) particle weights soft value estimate Doob h-transform on the discrete kernel Requires backprop ✓ ✗ ✗ ✗ Weighted selection unit — M particle trajectories M candidate states M candidate next states (same lx_l) Weight formula — path likelihood ratio estimated soft value transformed kernel via h^ω h^ω Doob kernel tilt ✗ ✗ ✗ ✓ Candidate diversity — different histories same state start same lx_l start Furthermore, we conducted detailed ablation studies on the D-DOIT algorithm using the regulatory DNA sequence design task to analyze the relationship between its computational efficiency and performance, and compared it with other algorithms (Appendices J and L). The task cell type specific enhancer design also features a detailed efficiency-performance analysis (Appendix K. Detailed experimental setups are also provided for the protein inverse folding task, serving as a complement to cross-modal transfer (Appendix M). Table 1 shows the differences between our algorithm and several other algorithms. 4.1 Task 1: Regulatory DNA Sequence Design Regulatory DNA elements direct cell-type-specific gene expression. They form an important target for gene therapy [24]. We use the HepG2 enhancer design benchmark of [8]: generate 200200 bp enhancer sequences with high predicted HepG2 activity and natural enhancer statistics on a frozen masked discrete diffusion backbone. The dataset, reward oracle, checkpoint, protocol, metrics, and baselines are documented in Sections J.1 and J.2. Table 2: HepG2 enhancer design results. We compare D-DOIT with training-free guidance and training-based finetuning baselines on DRAKES HepG2; all metrics are higher-is-better. Parentheses denote standard deviation across seeds, and bold/underline mark best/second best per column. D-DOIT uses held-out eval/test seed protocols in Sections J.6 and J.7. †SVDD-PM [12] is the matched-time control from Table 6 (M=416M=416), not part of the original DRAKES sweep. Method Pred-Activity (median) ↑ ATAC-Acc ↑ (%) 3-mer Corr ↑ JASPAR Corr ↑ Training-free methods Pre-trained 0.17(0.04) 1.5(0.2) -0.061(0.034) 0.249(0.015) CG 3.30(0.00) 0.0(0.0) -0.065(0.001) 0.212(0.035) SMC 4.15(0.33) 39.9(8.7) 0.840(0.045) 0.756(0.068) TDS 4.58(0.49) 40.0(14.1) 0.817(0.097) 0.798(0.044) SVDD-PM† 6.04(0.07) 39.8(0.4) 0.688(0.012) 0.917(0.002) D-DOIT-LBOK-S (Ours) 4.56(0.04) 65.0(1.4) 0.851(0.017) 0.948(0.002) D-DOIT-Q (Ours) 6.196.19(0.050.05) 91.7(0.9) 0.894(0.009) 0.923(0.002) Training-based methods CFG 5.04(0.06) 92.1(0.9) 0.746(0.001) 0.864(0.011) DRAKES w/o KL 6.44(0.04) 82.5(2.8) 0.307(0.001) 0.557(0.015) DRAKES 5.61(0.07) 92.5(0.6) 0.887(0.002) 0.911(0.002) Table 2 reports quality metrics. D-DOIT-Q outperforms every training-free baseline in Table 2 on every metric. D-DOIT-Q reaches 6.196.19 Pred-Activity, 91.7%91.7\% ATAC-Acc, 0.8940.894 3-mer Corr, and 0.9230.923 JASPAR Corr. It exceeds training-based DRAKES on Pred-Activity, 3-mer Corr, and JASPAR Corr, while remaining slightly lower on ATAC-Acc (91.7%91.7\% vs. 92.5%92.5\%). The near-TDS-speed D-DOIT-LBOK-S matches TDS on Pred-Activity (4.564.56 vs. 4.584.58) while lifting ATAC-Acc from 40.0%40.0\% to 65.0%65.0\%, 3-mer Corr from 0.8170.817 to 0.8510.851, and JASPAR Corr from 0.7980.798 to 0.9480.948 over TDS. At a similar measured wall-clock budget, Q improves all four metric means over SVDD-PM and improves activity, accessibility, and 3-mer correlation over Frozen Best-of-732; endpoint search retains higher JASPAR correlation. Sections J.3 and 6 gives the matched-time results and full protocol. A guided step costs 1+M1+M model forward passes plus M reward evaluations. The high-budget Q row is calibrated to endpoint sampling at approximately 10.610.6 seconds per output on the same A100; Appendix G gives the accounting. 4.2 Task 2: Cell-type-specific Enhancer Design Cell-type-specific enhancer design asks for sequences active in a target cell line and silent in off-target cell lines. This task is a stricter test of controllable generation than Section 4.1. We conduct experiments on the Ctrl-DNA enhancer benchmark of [25]. The benchmark spans three cell lines (HepG2, K562, SK-N-SH) and supplies cell-type-specific Enformer-based reward oracles. The dataset, composite specificity reward, per-cell-line oracles, diffusion training details, and the baseline-tuning protocol are documented in Sections K.1 and K.3. We report specificity (ΔR R), Motif Corr, and set-level diversity; ΔR R is the target-cell oracle score minus the average off-target score. Baselines include the training-free methods as in Section 4.1, regLM [26], RL baselines [25], and directed evolution; full metric definitions and baseline tuning are documented in the appendix, Sections K.2 and K.3. Table 3 reports three-cell-line averages. Among the diffusion-guidance baselines, D-DOIT clearly improves specificity: D-DOIT-LBOK-S raises ΔR R from 0.0020.002 to 0.0330.033 over TDS while improving Motif Corr from 0.5730.573 to 0.6480.648 at near-TDS wall-clock. D-DOIT-Q pushes ΔR R to 0.1190.119 while trading motif preservation for reward, giving the high-specificity point; LBOK-S remains the lower-budget point. Under the matched-time control, endpoint search also improves specificity and captures much of this gain, unlike the larger activity and accessibility separation on DRAKES. Q’s mean differences from Frozen Best-of-708 are small and are not claimed statistically significant, so the specificity gain is not unique to D-DOIT here. Wall-clock, per-cell, and diversity details and the matched-time control are in Sections K.4 and K.5. 4.3 Task 3: Protein Inverse Folding To test a different sequence space and reward, we apply D-DOIT to the DRAKES protein inverse-folding benchmark [8], using its reward oracle and evaluation metrics. Here the target backbone is fixed and the sampler generates amino-acid sequences. We evaluate predicted stability and agreement between the predicted structure and that backbone, with the full cross-method comparison including every reported metric for this benchmark in Table 3. Across all 1212 test backbones, three seeds, and 128128 sequences per backbone and seed, D-DOIT achieves joint success of 68.1±3.5%68.1± 3.5\%, compared with the published SMC and TDS results of 63.6%63.6\% and 62.9%62.9\% (Table 3). The configuration M=52M=52, K=8K=8, nh=1n_h=1 was fixed before test evaluation, with no post-test tuning, and each entry reports the mean and sample SD across the three seed-level metrics. D-DOIT has the highest predicted stability and joint success among the six methods, while TDS has better structural fidelity. The other five rows are taken from DRAKES Table 2, not rerun or compute-matched controls. Appendix M gives the full protocol. This supports transfer to a second biological sequence-design task, not experimental protein validation. Table 3: Cell-type-specific enhancer design (left) and protein inverse folding (right). Left: means over three cell lines, SD in parentheses; ΔR R is target minus average off-target reward; training-based references are in Tables 13 and K.4. Right: DRAKES protein benchmark; Pred-ddG and joint success are higher-is-better, scRMSD lower-is-better, SDs in Table 22. Bold marks the best mean per column, not significance. Method ΔR R ↑ Motif Corr. ↑ Training-free methods Pre-trained 0.000(0.003)0.000(0.003) 0.937(0.009)¯ 0.937(0.009) CG 0.000(0.003)0.000(0.003) 0.938(0.009)0.938(0.009) SMC 0.000(0.010)0.000(0.010) 0.587(0.114)0.587(0.114) TDS 0.002(0.009)0.002(0.009) 0.573(0.078)0.573(0.078) Frozen Best-of-708 0.108(0.028)0.108(0.028) 0.490(0.165)0.490(0.165) D-DOIT-LBOK-S 0.033(0.025)0.033(0.025) 0.648(0.046)0.648(0.046) D-DOIT-Q 0.119(0.063)¯ 0.119(0.063) 0.498(0.070)0.498(0.070) Method Pred-ddG ↑ scRMSD ↓ Success (%)↑\, Training-free methods Pre-trained −0.544-0.544 0.8490.849 34.434.4 CG −0.561-0.561 0.8390.839 34.734.7 SMC 0.6590.659 0.8410.841 63.663.6 TDS 0.6740.674 0.8340.834 62.962.9 D-DOIT (Ours) 0.8700.870 0.8720.872 68.168.1 Training-based methods CFG −1.186-1.186 3.1463.146 1.31.3 4.4 Compute–quality Trade-off To compare compute across methods, we account for oracle reward calls in addition to model evaluations. A guided update evaluates a shared proposal head, M candidate heads, and one reward per completed candidate; branching adds work only after the split. Table 4: Per-output DRAKES work. Full-window guidance, L=128L=128, one completion per candidate. Quantity LBOK-S Q (M,K,ssplit)(M,K,s_split) (9,4,105)(9,4,105) (52,8,0)(52,8,0) Model NFE 1,9701,970 54,27254,272 Candidate completions 1,7731,773 53,24853,248 The lower-budget DRAKES setting uses 1,9701,970 model evaluations and 1,7731,773 candidate completions per output, with 1,7771,777 reward evaluations after ranking the four final branches. Terminal ranking adds up to K queries when scores are not cached. In the 640640-output diagnostic, 1,2901,290 vectorized oracle calls score 1,137,2801,137,280 sequences, so physical calls and scored sequences are distinct cost units. Full accounting is given in Appendix G. Model NFE alone therefore does not determine total compute. In Table 7, Frozen Best-of-15 has a similar model budget (1,9201,920 versus 1,9701,970 evaluations per output) but only 1515 rather than 1,7771,777 oracle evaluations. Its 640640-output job is faster (97.897.8 versus 327.8327.8 seconds) but has lower Pred-Activity (2.2192.219 versus 4.4914.491). Thus, this comparison approximately matches model budget, not wall-clock time. The calibration in Table 6 instead matches endpoint sampling to Q at approximately 10.610.6 seconds per output. LBOK-S remains the near-TDS-runtime operating point (1.16×1.16× in the main timing study), whereas Q allocates substantially more compute to quality. For Ctrl-DNA, Q and Frozen Best-of-708 take 5.435.43 and 5.515.51 seconds per output on the same A100. Because absolute timings depend on batching and memory use, we report each study under its own timing protocol. D-DOIT generation costs recur for new samples, and comparisons with training-based methods concern sample quality rather than matched or amortized training cost. 4.5 Completion Quality and Search Diagnostics A crucial aspect of estimating h is how to obtain ^0 x_0; while a common approach to maintaining speed is to estimate ^0 x_0 directly from xtx_t in a single step, a more refined method involves obtaining ^0 x_0 through iterative steps starting from xtx_t. To better compare these two approaches, we investigated the performance of different estimation methods. Increasing completions per candidate from nh=1n_h=1 to 22 and 33 raises final guidance reward R from 5.272±0.0575.272± 0.057 to 5.764±0.0335.764± 0.033 and 5.978±0.0265.978± 0.026, at 41%41\% and 81%81\% additional wall time; we retain nh=1n_h=1 as the default. Direct and sequential completions have similar h-value distributions (Wasserstein-1 0.0600.060–0.1240.124; Section L.5), supporting practical use without exact-target equivalence. 4.6 Oracle Robustness The choice of reward function is independent of the D-DOIT algorithm itself, yet it significantly influences the algorithm’s performance. A more accurate reward function provides better guidance for the D-DOIT algorithm. To examine this, we designed various reward functions and observed their guiding effects. Replacing R with Enformer, CNN, or MLP predictions improves all four metrics over unguided sampling (Table 8); Pred-Activity reaches 4.4914.491, 2.2032.203, and 1.5441.544, versus 0.3710.371 for the frozen model. Held-out rescoring preserves these gains (Table 9), while evaluator scales remain incomparable. The controls support robustness across tested predictors without establishing experimental biological activity and are distinct from the high-budget 6.19±0.056.19± 0.05 result. 5 Conclusion and Discussion We introduced D-DOIT, a training-free sampler that applies Doob’s h-transform directly to frozen masked discrete diffusion models. Using a prediction-head h-estimator and finite-candidate Doob-weighted selection, D-DOIT supports black-box rewards without reward gradients or backbone retraining. Across DRAKES HepG2, Ctrl-DNA, and protein inverse folding, D-DOIT outperforms training-free guidance baselines, although its highest-quality results require a larger compute budget. The method has three main limitations. First, practical approximations, including temperature scaling, sharpening, and best-of-K selection, may introduce bias and do not retain the exact-target guarantee; late branching also cannot revise revealed tokens. Second, evaluation relies on predictive biological benchmarks and requires experimental validation. Third, the highest-quality setting incurs substantial recurring inference cost, leaving the reward–compute trade-off as a practical challenge. Overall, Doob-style kernel reweighting provides a general framework for guiding discrete generative models with sequence-level rewards. Acknowledgments 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 [1] J. Austin, D. D. Johnson, J. Ho, D. Tarlow, and R. van den Berg (2021) Structured denoising diffusion models in discrete state-spaces. In Advances in Neural Information Processing Systems, Vol. 34. External Links: Document Cited by: Appendix A, §1. [2] A. Campbell, J. Benton, V. De Bortoli, T. Rainforth, G. Deligiannidis, and A. Doucet (2022) A continuous time framework for discrete denoising models. Advances in Neural Information Processing Systems 35. External Links: Document Cited by: Appendix A, §1. [3] S. S. Sahoo, M. Arriola, Y. Schiff, A. Gokaslan, E. Marroquin, J. T. Chiu, A. Rush, and V. Kuleshov (2024) Simple and effective masked diffusion language models. In Advances in Neural Information Processing Systems, External Links: 2406.07524 Cited by: Appendix A, §J.1, §K.1, Appendix N, Appendix E, §1. [4] J. Shi, K. Han, Z. Wang, A. Doucet, and M. K. Titsias (2024) Simplified and generalized masked diffusion for discrete data. In Advances in Neural Information Processing Systems (NeurIPS), External Links: 2406.04329 Cited by: Appendix A, §1. [5] A. Lou, C. Meng, and S. Ermon (2024) Discrete diffusion modeling by estimating the ratios of the data distribution. In International Conference on Machine Learning (ICML), External Links: Document Cited by: Appendix A, §1. [6] H. Stark, B. Jing, C. Wang, G. Corso, B. Berger, R. Barzilay, and T. Jaakkola (2024) Dirichlet flow matching with applications to DNA sequence design. In International Conference on Machine Learning (ICML), External Links: Document Cited by: Appendix A, §1. [7] A. Sarkar, A. Duran, Y. Yu, D. Lin, Y. Kang, N. Somia, P. Mantilla, J. Zhou, M. Nagai, Z. Tang, K. Hanington, K. Chang, and P. K. Koo (2024) Designing DNA with tunable regulatory activity using discrete diffusion. bioRxiv. External Links: Document Cited by: Appendix A, §1. [8] C. Wang, M. Uehara, Y. He, A. Wang, A. Lal, T. Jaakkola, S. Levine, A. Regev, H. Wang, and T. Biancalani (2025) Fine-tuning discrete diffusion models via reward optimization with applications to DNA and protein design. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A, Appendix A, Appendix A, Appendix A, 1st item, 1st item, 1st item, §J.1, §J.1, §J.1, §J.1, Table 22, Appendix M, Appendix N, §1, §1, §4.1, §4.3. [9] J. Han, A. Wang, M. Xu, W. Chu, M. Dang, H. Ye, H. Chen, Y. Yue, and S. Ermon (2025) Discrete diffusion trajectory alignment via stepwise decomposition. arXiv preprint arXiv:2507.04832. External Links: 2507.04832 Cited by: Appendix A, §1. [10] H. Nisonoff, J. Xiong, S. Allenspach, and J. Listgarten (2025) Unlocking guidance for discrete state-space diffusion and flow models. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: 2nd item, 1st item, §1. [11] L. Wu, B. A. Trippe, C. A. Naesseth, D. M. Blei, and J. P. Cunningham (2023) Practical and asymptotically exact conditional sampling in diffusion models. In Advances in Neural Information Processing Systems, Vol. 36. External Links: 2306.17775 Cited by: Appendix A, 3rd item, 1st item, §1. [12] X. Li, Y. Zhao, C. Wang, G. Scalia, G. Eraslan, S. Nair, T. Biancalani, S. Ji, A. Regev, S. Levine, and M. Uehara (2025) Derivative-free guidance in continuous and discrete diffusion models with soft value-based decoding. In Advances in Neural Information Processing Systems, External Links: Document Cited by: Appendix A, Appendix A, 4th item, §1, Table 2. [13] M. Uehara, X. Su, Y. Zhao, X. Li, A. Regev, S. Ji, S. Levine, and T. Biancalani (2025) Reward-guided iterative refinement in diffusion models at test-time with applications to protein and DNA design. In Proceedings of the 42nd International Conference on Machine Learning, Vol. 267, p. 60515–60529. External Links: Document Cited by: Appendix A, §1. [14] Y. Schiff, S. S. Sahoo, H. Phung, G. Wang, S. Boshar, H. Dalla-Torre, B. P. de Almeida, A. Rush, T. Pierrot, and V. Kuleshov (2025) Simple guidance mechanisms for discrete diffusion models. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A, Appendix A, §1. [15] J. Rector-Brooks, M. Hasan, Z. Peng, C. Liu, S. Mittal, N. Dziri, M. Bronstein, P. Chatterjee, A. Tong, and J. Bose (2025) Steering masked discrete diffusion models via discrete denoising posterior prediction. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A, §1. [16] W. Guo, Y. Zhu, M. Tao, and Y. Chen (2024) Plug-and-play controllable generation for discrete masked models. arXiv preprint arXiv:2410.02143. External Links: Document Cited by: Appendix A, §1. [17] R. Singhal, Z. Horvitz, R. Teehan, M. Ren, Z. Yu, K. McKeown, and R. Ranganath (2025) A general framework for inference-time scaling and steering of diffusion models. In Proceedings of the 42nd International Conference on Machine Learning, Vol. 267, p. 55810–55827. External Links: 2501.06848 Cited by: Appendix A, §1. [18] T. Nguyen, K. Do, D. Kieu, and T. Nguyen (2025) H-edit: effective and flexible diffusion-based editing via doob’s h-transform. In Proceedings of the Computer Vision and Pattern Recognition Conference, p. 28490–28501. External Links: Document Cited by: §1. [19] Q. Zhu, Z. Ye, H. Liu, Z. Wang, and M. Chen (2026) Training-free adaptation of diffusion models via Doob’s h-transform. arXiv preprint arXiv:2602.16198. External Links: Document Cited by: Appendix A, Appendix C, Remark D.2, §1. [20] J. Heng, V. De Bortoli, A. Doucet, and J. Thornton (2025) Simulating diffusion bridges with score matching. Biometrika 112 (4), p. asaf048. External Links: Document Cited by: Appendix A, Appendix C, §1. [21] A. Denker, F. Vargas, S. Padhy, K. Didi, S. V. Mathis, R. Barbano, V. Dutordoir, E. Mathieu, U. J. Komorowska, and P. Lio (2024) DEFT: efficient fine-tuning of diffusion models by learning the generalised h-transform. In Advances in Neural Information Processing Systems (NeurIPS), External Links: Document Cited by: Appendix A, Appendix C, §1. [22] L. C. G. Rogers and D. Williams (2000) Diffusions, markov processes, and martingales. Vol. 2, Cambridge university press. External Links: Document Cited by: §2. [23] S. Särkkä and A. Solin (2019) Applied stochastic differential equations. Vol. 10, Cambridge University Press. External Links: Document Cited by: §2. [24] S. J. Gosai, R. I. Castro, N. Fuentes, J. C. Butts, K. Mouri, M. Alasoadura, S. Kales, T. T. L. Nguyen, R. R. Noche, A. S. Rao, M. T. Joy, P. C. Sabeti, S. K. Reilly, and R. Tewhey (2024) Machine-guided design of cell-type-targeting cis-regulatory elements. Nature 634, p. 1211–1220. External Links: Document Cited by: Appendix A, 3rd item, §J.1, Appendix N, §4.1. [25] X. Chen, S. Ma, R. Lin, J. Lin, and B. Wang (2025) Ctrl-DNA: controllable cell-type-specific regulatory DNA design via constrained RL. arXiv preprint arXiv:2505.20578. External Links: Document Cited by: Appendix A, Appendix A, Figure 3, 2nd item, 3rd item, 3rd item, 4th item, §K.1, §K.1, §K.1, §K.1, §K.1, §K.2, §K.2, Table 14, Appendix N, §4.2, §4.2. [26] A. Lal, D. Garfield, T. Biancalani, and G. Eraslan (2024) RegLM: designing realistic regulatory DNA with autoregressive language models. bioRxiv, p. 2024–02. External Links: Document Cited by: Appendix A, 2nd item, §K.1, §4.2. [27] D. Guo, D. Yang, H. Zhang, J. Song, P. Wang, Q. Zhu, R. Xu, R. Zhang, S. Ma, X. Bi, et al. (2025) Deepseek-r1: incentivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948. Cited by: Appendix A. [28] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al. (2020) Language models are few-shot learners. Advances in neural information processing systems 33, p. 1877–1901. Cited by: Appendix A. [29] Z. Wang, Y. Lu, Y. Zhang, Q. Liu, C. Luo, J. Gesi, H. Lu, Y. Sang, M. Li, J. Huang, et al. (2026) SENTINEL: failure-driven reinforcement learning for training tool-using language model agents. arXiv preprint arXiv:2606.12908. Cited by: Appendix A. [30] Z. Wang, Y. Lu, W. Li, A. Amini, B. Sun, Y. Bart, W. Lyu, J. Gesi, T. Wang, J. Huang, et al. (2026) Opera: a dataset of observation, persona, rationale, and action for evaluating llms on human online shopping behavior simulation. In Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), p. 43942–43960. Cited by: Appendix A. [31] Z. Wang, Y. Lu, Y. Zhang, J. Huang, and D. Wang (2025) Customer-r1: personalized simulation of human behaviors via rl-based llm agent in online shopping. arXiv preprint arXiv:2510.07230. Cited by: Appendix A. [32] Z. Wang, Y. Lu, Y. Zhang, P. Chen, Z. Dong, J. Huang, J. Gesi, X. Tang, C. Luo, Q. Liu, et al. (2026) Trajectory2task: training robust tool-calling agents with synthesized yet verifiable data for complex user intents. In Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), p. 44021–44044. Cited by: Appendix A. [33] J. Ho, A. Jain, and P. Abbeel (2020) Denoising diffusion probabilistic models. Advances in neural information processing systems 33, p. 6840–6851. Cited by: Appendix A. [34] J. Y. Hu, W. Wu, Z. Li, S. Pi, Z. Song, and H. Liu (2024) On statistical rates and provably efficient criteria of latent diffusion transformers (dits). Advances in Neural Information Processing Systems 37, p. 31562–31628. Cited by: Appendix A. [35] J. Y. Hu, W. Wu, Y. Lee, Y. Huang, M. Chen, and H. Liu (2025) On statistical rates of conditional diffusion transformers: approximation, estimation and minimax optimality. In International Conference on Learning Representations, Vol. 2025, p. 12591–12675. Cited by: Appendix A. [36] E. Hoogeboom, D. Nielsen, P. Jaini, P. Forré, and M. Welling (2021) Argmax flows and multinomial diffusion: learning categorical distributions. In Advances in Neural Information Processing Systems (NeurIPS), External Links: Document Cited by: Appendix A. [37] J. Ou, S. Nie, K. Xue, F. Zhu, J. Sun, Z. Li, and C. Li (2025) Your absorbing discrete diffusion secretly models the conditional distributions of clean data. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A. [38] A. Campbell, J. Yim, R. Barzilay, T. Rainforth, and T. Jaakkola (2024) Generative flows on discrete state-spaces: enabling multimodal flows with applications to protein co-design. In International Conference on Machine Learning (ICML), External Links: Document Cited by: Appendix A. [39] I. Gat, T. Remez, N. Shaul, F. Kreuk, R. T. Q. Chen, G. Synnaeve, Y. Adi, and Y. Lipman (2024) Discrete flow matching. In Advances in Neural Information Processing Systems (NeurIPS), External Links: 2407.15595 Cited by: Appendix A. [40] K. Black, M. Janner, Y. Du, I. Kostrikov, and S. Levine (2024) Training diffusion models with reinforcement learning. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A. [41] Y. Fan, O. Watkins, Y. Du, H. Liu, M. Ryu, C. Boutilier, P. Abbeel, M. Ghavamzadeh, K. Lee, and K. Lee (2023) DPOK: reinforcement learning for fine-tuning text-to-image diffusion models. In Advances in Neural Information Processing Systems (NeurIPS), External Links: 2305.16381 Cited by: Appendix A. [42] K. Clark, P. Vicol, K. Swersky, and D. J. Fleet (2024) Directly fine-tuning diffusion models on differentiable rewards. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A. [43] C. Domingo-Enrich, M. Drozdzal, B. Karrer, and R. T. Q. Chen (2025) Adjoint matching: fine-tuning flow and diffusion generative models with memoryless stochastic optimal control. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A. [44] B. L. Trippe, J. Yim, D. Tischer, D. Baker, T. Broderick, R. Barzilay, and T. Jaakkola (2023) Diffusion probabilistic modeling of protein backbones in 3d for the motif-scaffolding problem. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A. [45] G. Cardoso, S. Le Corff, and E. Moulines (2024) Monte carlo guided diffusion for bayesian linear inverse problems. In International Conference on Learning Representations (ICLR), External Links: 2308.07983 Cited by: Appendix A. [46] H. Chung, J. Kim, M. T. Mccann, M. L. Klasky, and J. C. Ye (2023) Diffusion posterior sampling for general noisy inverse problems. In International Conference on Learning Representations (ICLR), External Links: Document Cited by: Appendix A. [47] P. Phunyaphibarn and M. Sung (2026) Reward-guided discrete diffusion via clean-sample markov chain for molecule and biological sequence design. arXiv preprint arXiv:2602.09424. External Links: Document, 2602.09424, Link Cited by: Appendix A, Appendix A. [48] N. Ma, S. Tong, H. Jia, H. Hu, Y. Su, M. Zhang, X. Yang, Y. Li, T. Jaakkola, X. Jia, and S. Xie (2025) Inference-time scaling for diffusion models beyond scaling denoising steps. arXiv preprint arXiv:2501.09732. External Links: Document Cited by: Appendix A. [49] J. L. Doob (1984) Classical potential theory and its probabilistic counterpart. Springer, Berlin. External Links: Document Cited by: Appendix A. [50] V. De Bortoli, J. Thornton, J. Heng, and A. Doucet (2021) Diffusion schrödinger bridge with applications to score-based generative modeling. In Advances in Neural Information Processing Systems (NeurIPS), External Links: 2106.01357 Cited by: Appendix A. [51] J. Kim, T. Yoon, P. Phunyaphibarn, S. Kim, M. Mardani, and M. Sung (2026) Contrastive distribution matching for amortized sequential monte carlo in discrete diffusion. arXiv preprint arXiv:2605.23346. External Links: 2605.23346, Link Cited by: Appendix A. [52] D. Bu, W. Huang, A. Han, S. Wu, H. Wong, Q. Zhang, T. Suzuki, and A. Nitanda (2026) DPRM: a plug-in doob h transform-induced token-ordering module for discrete diffusion models. arXiv preprint arXiv:2604.24357. External Links: 2604.24357, Link Cited by: Appendix A. [53] I. I. Taskiran, K. I. Spanier, H. Dickmanken, N. Kempynck, A. Pančíková, E. C. Eksşi, G. Hulselmans, J. N. Ismail, K. Theunis, R. Vandepoel, et al. (2024) Cell-type-directed design of synthetic enhancers. Nature 626 (7997), p. 212–220. External Links: Document Cited by: Appendix A. [54] M. Su, P. Hsieh, W. Wu, M. Lu, J. Chen, J. Y. Hu, and H. Liu (2026) Discrete flow matching policy optimization. arXiv preprint arXiv:2604.06491. Cited by: Appendix A. [55] E. Nguyen, M. Poli, M. Faizi, A. Thomas, C. Birch-Sykes, M. Wornow, A. Patel, C. Rabideau, S. Massaroli, Y. Bengio, et al. (2023) HyenaDNA: long-range genomic sequence modeling at single nucleotide resolution. Advances in Neural Information Processing Systems. External Links: 2306.15794 Cited by: Appendix A, Appendix N. [56] Y. Schiff, C. Kao, A. Gokaslan, T. Dao, A. Gu, and V. Kuleshov (2024) Caduceus: bi-directional equivariant long-range dna sequence modeling. In International Conference on Machine Learning (ICML), External Links: Document Cited by: Appendix A. [57] H. Dalla-Torre, L. Gonzalez, J. Mendoza-Revilla, N. Lopez Carranza, A. H. Grzywaczewski, F. Oteri, C. Dallago, E. Trop, B. P. de Almeida, H. Sirelkhatim, et al. (2024) Nucleotide transformer: building and evaluating robust foundation models for human genomics. Nature Methods. External Links: Document Cited by: Appendix A. [58] Z. Zhou, R. Riley, S. Kautsar, W. Wu, R. Egan, S. Hofmeyr, S. Goldhaber-Gordon, M. Yu, H. Ho, F. Liu, et al. (2025) Genomeocean: an efficient genome foundation model trained on large-scale metagenomic assemblies. bioRxiv. Cited by: Appendix A. [59] W. Wu, X. Song, Y. Wen, Q. Lin, Z. Zhou, J. Y. Hu, Z. Wang, and H. Liu (2025) Genome-factory: a library for tuning, deploying, and interpreting genomic foundation models. arXiv preprint arXiv:2509.12266. Cited by: Appendix A. [60] Z. Zhou, W. Wu, H. Ho, J. Wang, L. Shi, R. V. Davuluri, Z. Wang, and H. Liu (2025) DNABERT-s: pioneering species differentiation with species-aware dna embeddings. Bioinformatics 41 (Supplement_1), p. i255–i264. Cited by: Appendix A. [61] P. Avdeyev, C. Shi, Y. Tan, K. Dudnyk, and J. Zhou (2023) Dirichlet diffusion score model for biological sequence generation. In International Conference on Machine Learning (ICML), External Links: Document Cited by: Appendix A. [62] A. Awasthi, R. Bednarsky, M. Schaefer, and C. Bock (2026) Conditional monte carlo tree diffusion for designing cell-type-specific and biologically faithful regulatory DNA. arXiv preprint arXiv:2604.20488. External Links: Document Cited by: Appendix A, Appendix A. [63] B. D. O. Anderson (1982) Reverse-time diffusion equation models. Stochastic Processes and their Applications 12 (3), p. 313–326. External Links: Document Cited by: Appendix C. [64] Ž. Avsec, V. Agarwal, D. Visentin, J. R. Ledsam, A. Grabska-Barwinska, K. R. Taylor, Y. Assael, J. Jumper, P. Kohli, and D. R. Kelley (2021) Effective gene expression prediction from sequence by integrating long-range interactions. Nature Methods 18 (10), p. 1196–1203. External Links: Document Cited by: §J.1, §K.1, Appendix N. [65] ENCODE Project Consortium (2012) An integrated encyclopedia of DNA elements in the human genome. Nature 489 (7414), p. 57–74. External Links: Document Cited by: 2nd item, Appendix N. [66] J. A. Castro-Mondragón, R. Riudavets-Puig, I. Rauluseviciute, R. B. Lemma, L. Turchi, R. Blanc-Mathieu, J. Lucas, P. Boddie, A. Khan, N. Manosalva Pérez, et al. (2022) JASPAR 2022: the 9th release of the open-access database of transcription factor binding profiles. Nucleic Acids Research 50 (D1), p. D165–D173. External Links: Document Cited by: 4th item, Appendix N. [67] J. Ho and T. Salimans (2022) Classifier-free diffusion guidance. arXiv preprint arXiv:2207.12598. External Links: Document Cited by: 3rd item. Appendix Appendix A Related Work We review related work on reward-guided adaptation of discrete diffusion and flow models, as well as regulatory DNA sequence design. Reward-guided adaptation of discrete diffusion and flow models. Reward-guided generation has proved effective for aligning language models [27, 28, 29, 30, 31, 32] and diffusion models [33, 34, 35] with task-specific objectives. In discrete generative models, however, guidance must act on categorical transitions over a finite vocabulary. Early discrete denoising diffusion models use structured categorical or absorbing-state corruption [1, 36, 2]; masked discrete diffusion models scale a simple absorbing-mask objective [3, 4]; score-entropy and ratio-matching parameterizations form a non-masked alternative [5, 37]; and discrete flow matching extends the same idea to a flow framework [38, 39]. Reward adaptation for these models can be training-based or training-free. Training-based methods include DRAKES, which finetunes the generator through reward backpropagation [8], trajectory-level alignment [9], and reinforcement learning (RL)-style finetuning methods developed for continuous diffusion [40, 41, 42, 43]; these methods change backbone weights and need a differentiable reward. Training-free methods instead steer a frozen backbone at inference time, including Sequential Monte Carlo and twisted sampling [11, 44, 45], posterior sampling for inverse problems [46], soft value-based decoding [12], reward-guided iterative refinement [13], clean-sample Markov chains [47], simple guidance for masked discrete diffusion [14], denoising posterior prediction [15], plug-and-play control [16], and inference-time scaling [17, 48]. These methods steer a frozen backbone, but they do not directly tilt the discrete reverse kernel toward an optimal proposal. Comparing with three representative efforts highlights the distinct role of D-DOIT. DRAKES [8] provides a strong reward-optimization baseline, but it finetunes the backbone and requires a differentiable reward. Soft value-based decoding [12] selects candidates using importance scores on intermediate latents, but it does not tilt the reverse transition kernel itself. Simple masked-diffusion guidance [14] introduces a heuristic guidance term, but lacks an explicit optimality target. In contrast, D-DOIT is the first inference-time method to implement an executable estimator of Doob’s h-transform for the discrete reverse kernel [49, 19, 20, 21, 50]. It scores candidates through the resulting h-tilted kernel, combines a frozen masked discrete diffusion backbone with a black-box reward, and further supports optional best-of-K refinement over the same frozen prior. Because finite-candidate value-weighted selection also appears in soft value-based decoding and related methods, we do not claim that local candidate selection is unique to D-DOIT. CDM [51] learns a twist to amortize value estimation, and DPRM [52] applies Doob-derived corrections to token ordering while retaining a host token sampler; CSMC [47] instead searches over clean samples. Our distinction is the explicit categorical reverse-kernel h-transform as the reference object, evaluated with a frozen prediction-head proxy rather than a learned value network. Regulatory DNA sequence design. Regulatory DNA design aims to generate sequences that score well under predictive assays and stay faithful to natural regulatory grammar. Oracle-guided optimization with experimental validation drives early machine-guided enhancer design [53, 24, 54]; autoregressive regulatory language models [26] and genomic foundation models [55, 56, 57, 58, 59, 60] provide strong sequence priors; and diffusion-based generators specialize these priors through Dirichlet diffusion and score-entropy variants [6, 7, 61]. For reward-optimization benchmarks, DRAKES evaluates HepG2 enhancer generation through activity, accessibility, k-mer, and motif-structure metrics [8]; Ctrl-DNA studies cell-type-specific enhancer design through constrained reinforcement learning [25]; and concurrent work combines discrete diffusion with tree search for cell-type-specific regulatory DNA design under related biological reward objectives [62]. The strongest methods on these benchmarks share one limit: they reach high reward only through backbone updates, policy training, or expanded inference search. Comparing with three representative efforts highlights the efficiency and generality of D-DOIT. DRAKES [8] finetunes the backbone and requires a differentiable reward. Ctrl-DNA [25] achieves cell-type specificity through constrained RL with reward signals shaped for policy optimization. The concurrent tree-search method of Awasthi et al. [62] remains training-free at inference time, but expands a search tree over partial sequences and increases the per-step sampling cost. In contrast, D-DOIT is the first method to compete on both benchmarks without policy training. It trades modest inference-time compute for reward gains while preserving near-base per-step cost. Appendix B Proof of Theorem 2.1 Proof. Let ϕ(0)=exp(R(0)/β),Zβ=pθ[ϕ(0)]. φ(x_0)= (R( _0)/β), Z_β=E_p_θ [φ(X_0) ]. All identities below are understood on the support of the pretrained reverse chain. Let ℙθP_θ denote the path law of the pretrained reverse chain, ℙθ(¯t0,…,¯tL)=pθ,tL(¯tL)∏l=1LPθ,tl(¯tl−1∣¯tl). _θ( x_t_0,…, x_t_L)=p_θ,t_L( x_t_L) _l=1^LP_θ,t_l( x_t_l-1 x_t_l). Define the reward-tilted path law by dℙθhdℙθ(¯t0,…,¯tL)=ϕ(0)Zβ. _θ^h _θ( X_t_0,…, X_t_L)= φ(X_0)Z_β. We now compute the one-step reverse transition under this tilted path law. For l=1,…,Ll=1,…,L, ℙθh(¯tl−1=¯tl−1∣¯tl=¯tl) _θ^h ( X_t_l-1= x_t_l-1 X_t_l= x_t_l ) =pθ[¯tl−1=¯tl−1ϕ(0)∣¯tl=¯tl]pθ[ϕ(0)∣¯tl=¯tl] = E_p_θ [1\ X_t_l-1= x_t_l-1\φ(X_0) X_t_l= x_t_l ]E_p_θ [φ(X_0) X_t_l= x_t_l ] =Pθ,tl(¯tl−1∣¯tl)pθ[ϕ(0)∣¯tl−1=¯tl−1,¯tl=¯tl]h(¯tl,tl). =P_θ,t_l ( x_t_l-1 x_t_l ) E_p_θ [φ(X_0) X_t_l-1= x_t_l-1, X_t_l= x_t_l ]h( x_t_l,t_l). By the Markov property of the reverse chain, once ¯tl−1 X_t_l-1 is given, the future evolution toward clean time is independent of ¯tl X_t_l. Hence pθ[ϕ(0)∣¯tl−1=¯tl−1,¯tl=¯tl]=h(¯tl−1,tl−1). _p_θ [φ(X_0) X_t_l-1= x_t_l-1, X_t_l= x_t_l ]=h( x_t_l-1,t_l-1). Therefore ℙθh(¯tl−1=¯tl−1∣¯tl=¯tl)=Pθ,tl(¯tl−1∣¯tl)h(¯tl−1,tl−1)h(¯tl,tl). _θ^h ( X_t_l-1= x_t_l-1 X_t_l= x_t_l )=P_θ,t_l ( x_t_l-1 x_t_l ) h( x_t_l-1,t_l-1)h( x_t_l,t_l). This proves that the reward-tilted path law induces the Doob-transformed reverse kernel in (2.5). In particular, the kernel is normalized because ∑¯tl−1Pθ,tl(¯tl−1∣¯tl)h(¯tl−1,tl−1)h(¯tl,tl)=pθ[h(¯tl−1,tl−1)∣¯tl=¯tl]h(¯tl,tl)=1, _ x_t_l-1P_θ,t_l ( x_t_l-1 x_t_l ) h( x_t_l-1,t_l-1)h( x_t_l,t_l)= E_p_θ [h( X_t_l-1,t_l-1) X_t_l= x_t_l ]h( x_t_l,t_l)=1, where the last equality follows from the tower property and the definition of h. It remains to compute the marginals. For any l=0,…,Ll=0,…,L, pθ,tlh(¯tl) p_θ,t_l^h( x_t_l) =ℙθh(¯tl=¯tl) =P_θ^h( X_t_l= x_t_l) =pθ[¯tl=¯tlϕ(0)]Zβ = E_p_θ [1\ X_t_l= x_t_l\φ(X_0) ]Z_β =pθ,tl(¯tl)pθ[ϕ(0)∣¯tl=¯tl]Zβ = p_θ,t_l( x_t_l)E_p_θ [φ(X_0) X_t_l= x_t_l ]Z_β =pθ,tl(¯tl)h(¯tl,tl)Zβ. = p_θ,t_l( x_t_l)h( x_t_l,t_l)Z_β. In the fully masked terminal setting, ¯tL X_t_L is deterministic. Hence h(¯tL,tL)=pθ[ϕ(0)]=Zβ, h( X_t_L,t_L)=E_p_θ [φ(X_0) ]=Z_β, so the transformed terminal distribution coincides with the pretrained terminal distribution. Finally, since ¯t0=0 X_t_0=X_0, h(0,0)=pθ[ϕ(0)∣0=0]=ϕ(0)=exp(R(0)/β). h(x_0,0)=E_p_θ [φ(X_0) _0=x_0 ]=φ(x_0)= (R( _0)/β). Therefore pθ,0h(0)=pθ,0(0)exp(R(0)/β)Zβ=πβ(0). p_θ,0^h(x_0)= p_θ,0(x_0) (R( _0)/β)Z_β= _β(x_0). This completes the proof. ∎ A separate tempered reference chain. The theorem above retains the original frozen PθP_θ reference and target πβ _β. The following proposition describes a different reference law; it neither replaces that target nor identifies the executed estimator with an exact Doob value. Proposition B.1 (Doob h-transform relative to a tempered reference). Fix τ>0τ>0. Let QlτQ^τ_l be the reverse kernel obtained from Pθ,tlP_θ,t_l by replacing μθ _θ with the normalized μθ,τ _θ,τ in (3.6), while preserving the reveal schedule and copy rule. Let ℚτQ^τ be the path law using these kernels at every reverse step and starting from the deterministic all-mask state ¯tL=N x_t_L= m^N. Assume the chain ends in X, and let q0τq^τ_0 be its clean-time law. For η>0η>0, suppose 0<Zτ,η:=ℚτ[exp(ηR(0))]<∞0<Z_τ,η:=E_Q^τ[ (η R( _0))]<∞. Define the true reference-chain value, on reachable states, by hτ,η(¯,tl) h_τ,η( x,t_l) =ℚτ[exp(ηR(0))∣¯tl=¯]. =E_Q^τ[ (η R( _0)) X_t_l= x]. Then the kernels Qlτ,h(¯∣¯) Q^τ,h_l( y x) =Qlτ(¯∣¯)hτ,η(¯,tl−1)hτ,η(¯,tl) =Q^τ_l( y x) h_τ,η( y,t_l-1)h_τ,η( x,t_l) are normalized. Starting from the same deterministic mask, their clean-time marginal is q0τ,h(0) q^τ,h_0(x_0) =q0τ(0)exp(ηR(0))Zτ,η. = q^τ_0(x_0) (η R( _0))Z_τ,η. Proof. The Markov and tower properties give hτ,η(¯,tl) h_τ,η( x,t_l) =∑¯Qlτ(¯∣¯)hτ,η(¯,tl−1), = _ yQ^τ_l( y x)h_τ,η( y,t_l-1), which proves normalization and positivity on the reference support. At clean time hτ,η(0,0)=exp(ηR(0))h_τ,η(x_0,0)= (η R( _0)); at the deterministic initial mask hτ,η(N,tL)=Zτ,ηh_τ,η( m^N,t_L)=Z_τ,η. Consequently, the likelihood ratio of the transformed path law to ℚτQ^τ telescopes to ∏l=1Lhτ,η(¯tl−1,tl−1)hτ,η(¯tl,tl) _l=1^L h_τ,η( x_t_l-1,t_l-1)h_τ,η( x_t_l,t_l) =exp(ηR(0))Zτ,η. = (η R( _0))Z_τ,η. Marginalizing this identity over the intermediate states yields the claimed clean-time law. ∎ For τ=1τ=1, η=1/βη=1/β this recovers the P-reference theorem. Otherwise the target changes; the practical proxy, windowing, and terminal selection do not inherit its exactness. Appendix C Continuous-time Doob’s h-transform on Score-based Diffusion We record the standard continuous-time derivation for comparison with the discrete construction. Consider a forward score-based diffusion on ℝdR^d, dt=f(t,t)dt+g(t)dt,t∈[0,T], _t=f(X_t,t)\, t+g(t)\, _t, t∈[0,T], with marginal density ptp_t and 0∼p0X_0 p_0. The reverse-time process admits the well-known form dt=(f(t,t)−g(t)2∇logpt(t))dt+g(t)d¯t, _t= (f(X_t,t)-g(t)^2∇ p_t(X_t) )\, t+g(t)\, W_t, where ¯t W_t is reverse-time Brownian motion [63]. For the continuous analogue of the reward-tilted target in (2.3), write the base terminal density as p0p_0 and denote the target by π(0)∝p0(0)ϕ(0)π(x_0) p_0(x_0)φ(x_0). Define the h-function ht()=[ϕ(0)|t=]. h_t(x)=E [φ(X_0)\, |\,X_t=x ]. Doob’s h-transform replaces ptp_t with the tilted marginal p~t∝ptht p_t p_th_t, which gives the tilted reverse drift (f(,t)−g(t)2∇logpt())−g(t)2∇loght()=f(,t)−g(t)2∇log(pt()ht()). (f(x,t)-g(t)^2∇ p_t(x) )-g(t)^2∇ h_t(x)=f(x,t)-g(t)^2∇ (p_t(x)h_t(x) ). Equivalently, the score is shifted by an additive correction: ∇logpt()⟶∇logpt()+∇loght(). ∇ p_t(x)\; \;∇ p_t(x)+∇ h_t(x). With initialization from the tilted marginal at time T, the terminal law of the tilted reverse process is π [19, 20, 21]; using the base marginal at T is exact only when that marginal is unchanged by the tilt. The continuous construction requires sufficient regularity of hth_t for ∇loght∇ h_t to exist, not differentiability of R itself. Under suitable smoothing and integrability conditions, differentiating a transition density can give this correction even for a black-box reward. The structural distinction for masked discrete diffusion is instead that ~ X is categorical and has no Euclidean score field. Section 3 therefore formulates a discrete kernel-level analogue, where the tilt enters as a multiplicative reweighting of categorical transition probabilities. Appendix D On the Prediction-head h-estimator Definition 3.1 is the executable counterpart of the rollout h-function used by the exact discrete transform. For attention-based masked discrete diffusion models, it is generally an estimator rather than an exact identity. For a candidate state ¯tl−1 y_t_l-1 at time tl−1t_l-1, the exact quantity is h(¯tl−1,tl−1)=pθ[ϕ(0)∣¯tl−1=¯tl−1], h( y_t_l-1,t_l-1)=E_p_θ[φ(X_0) X_t_l-1= y_t_l-1], where the expectation runs over the pre-trained reverse chain from time tl−1t_l-1 down to clean time, conditional on ¯tl−1=¯tl−1 X_t_l-1= y_t_l-1. At each subsequent reverse step l′∈l−1,l−2,…,1l ∈\l-1,l-2,…,1\, the reverse kernel (2.1) reveals each still-masked position from μθj(x0j∣¯tl′,tl′) _θ^j(x_0^j x_t_l ,t_l ) and treats positions as conditionally independent. The copy flag freezes any position that the chain has already revealed. The exact rollout marginal of 0X_0 given ¯tl−1=¯tl−1 X_t_l-1= y_t_l-1 therefore mixes per-position predictions at the intermediate states ¯tl′ x_t_l that the chain visits, which carry more revealed context than ¯tl−1 y_t_l-1. The factorized head (3.1) evaluates the clean completion from the current state ¯tl−1 y_t_l-1 itself, giving pθhead(0∣¯tl−1)=∏j:y¯tl−1j=μθj(x0j∣¯tl−1,tl−1)⋅∏j:y¯tl−1j≠δy¯tl−1j(x0j). p^head_θ(x_0 y_t_l-1)= _j:\, y_t_l-1^j= m _θ^j(x_0^j y_t_l-1,t_l-1)· _j:\, y_t_l-1^j≠ m _ y_t_l-1^j(x_0^j). (D.1) A sufficient condition for this distribution to coincide with the full rollout marginal is that each still-masked coordinate’s head prediction remains invariant across all reachable subsequent contexts and times until that coordinate is revealed. For attention-based masked discrete diffusion models this invariance need not hold. The direct completion law therefore generally differs from the multi-step rollout law, even with arbitrarily many completion samples. To separate sampling noise from this distributional mismatch, let nhn_h denote the number of independent completions of a fixed candidate (reserving N for sequence length), as in (3.5), and write h^nh(¯,t) h_n_h( y,t) =1nh∑i=1nhexp(R(^0(i))/β),^0(i)∼pθ,γhead(⋅∣¯), = 1n_h _i=1^n_h (R( _0^(i))/β), x_0^(i) p^head_θ,γ(· y), h¯θ,γ(¯,t) h_θ,γ( y,t) =pheadθ,γ(⋅∣¯)[exp(R(0)/β)]. =E_p^head_θ,γ(· y)[ (R( _0)/β)]. The executed single-completion estimator has nh=1n_h=1. As nh→∞n_h→∞, the sample average converges to h¯θ,γ h_θ,γ, not generally to the rollout value h; at γ=1γ=1 it recovers the head expectation in (3.2). Remark D.1 (Practical tempering and selection bias). Completion temperature γ≠1γ≠ 1 changes the direct completion law and hence its reward expectation; it does not merely reduce Monte Carlo noise. Proposal temperature τ≠1τ≠ 1 changes the candidate kernel, denoted QlτQ^τ_l, relative to Pθ,tlP_θ,t_l. With independent candidate draws and completion randomness, the large-M limit of the uncorrected selection rule at a fixed current state ¯ x is proportional to Qlτ(¯∣¯)[h^nh(¯,tl−1)ω∣¯]. Q^τ_l( y x)\,E[ h_n_h( y,t_l-1)^ω y]. On the proposal support, multiplying each weight by Pθ,tl(¯∣¯)/Qlτ(¯∣¯)P_θ,t_l( y x)/Q^τ_l( y x) as in (3.8) corrects only this proposal mismatch, provided the proposal covers the base-kernel support. It does not correct the head-versus-rollout mismatch, completion tempering, nonlinear sharpening, or finite-M self-normalization. The executed sampler omits this likelihood ratio; its uncorrected tempered update is not the exact PθP_θ transform. In particular, [h^nhω∣¯]E[ h_n_h^ω y] generally differs from ([h^nh∣¯])ω(E[ h_n_h y])^ω and from h(¯,t)ωh( y,t)^ω. Even with exact rollout values, raising P[exp(R/β)∣¯]E_P[ (R/β) y] to ω is generally not the same as computing P[exp(ωR/β)∣¯]E_P[ (ω R/β) y]. For nh=1n_h=1, sharpening does yield an expectation of exp(ωR/β) (ω R/β) under the direct completion law, but that law is still not the rollout law. A separate exact transform relative to a tempered reference chain is stated in Appendix B; it requires that chain’s true downstream expectation, not the practical estimator. Finally, restricting guidance to a window omits exact transformed updates outside that window, and terminal best-of-K ranking generally changes the output law even if each branch were sampled exactly. Thus these operating choices are not a guarantee of sampling (2.3); their empirical effects are studied in Section L.3. Remark D.2 (Comparison with the continuous case). The continuous-time h-function identity in [19] differentiates the Gaussian transition density to bypass the requirement that ϕφ be differentiable. The discrete construction avoids differentiating through ϕφ for a different reason: we query ϕφ only on a sampled clean completion ^0∼pθ,γhead(⋅∣¯tl−1) x_0 p^head_θ,γ(· y_t_l-1), so a single forward pass through the prediction head suffices to produce h^(¯tl−1,tl−1)=ϕ(^0) h( y_t_l-1,t_l-1)=φ( x_0) as in (3.3). The trade-off is that this single-shot estimator uses the factorized prediction head (3.1) instead of explicitly simulating the multi-step rollout marginal under the frozen reverse chain. Appendix E Standard Masked Discrete Diffusion Sampling Algorithm 1 records the standard generative process of the pre-trained masked discrete diffusion model [3] that D-DOIT builds on, provided here as background. The forward process corrupts tokens with an absorbing m state. The reverse process evaluates the clean-sequence prediction head pθhead(0∣¯tl)p^head_θ(x_0 x_t_l) at each step. It then unmasks each still-masked position with the per-step reveal schedule from (2.1) and treats positions as conditionally independent. Once a step reveals a token, later steps freeze it. For consistency with the main text, Algorithm 1 writes a linear schedule with αtl=1−l/L _t_l=1-l/L; other monotone schedules can be substituted. Algorithm 1 Standard Masked Discrete Diffusion Sampling (Baseline) 0: Pre-trained masked discrete diffusion model pθp_θ, steps L 1: Initialize ¯tL←(,…,) x_t_L←( m,…, m) all positions masked 2: for l=L,L−1,…,1l=L,L-1,…,1 do 3: Set αtl=1−l/L _t_l=1-l/L, αtl−1=1−(l−1)/L _t_l-1=1-(l-1)/L linear schedule for this baseline 4: Forward pass: obtain μθj(v∣¯tl,tl) _θ^j(v x_t_l,t_l) for all positions j one model forward pass 5: for each masked position j (where x¯tlj= x_t_l^j= m) do 6: Sample x¯tl−1j x_t_l-1^j: unmask to token v∈v with prob μθj(v∣¯tl,tl)(αtl−1−αtl)/(1−αtl) _θ^j(v x_t_l,t_l)( _t_l-1- _t_l)/(1- _t_l); stay masked with prob (1−αtl−1)/(1−αtl)(1- _t_l-1)/(1- _t_l) (2.1) 7: end for 8: Copy flag: x¯tl−1j←x¯tlj x_t_l-1^j← x_t_l^j for all unmasked j revealed tokens are frozen 9: end for 10: return ¯0 x_0 Appendix F Masked Diffusion Transition Details This section records the explicit matrices behind the forward and reverse kernels summarized in Section 2.1. For x∈~x∈ V, let ex∈ℝd+1e_x ^d+1 be its one-hot vector including e_ m for the mask token, and let ∈ℝd+11 ^d+1 be the all-one vector. Let (Xtl)l=0L(X_t_l)_l=0^L be the single-token chain with X0∈X_0 . Under the column-vector convention, the forward transition matrix on [tl−1,tl][t_l-1,t_l] is Kl=(1−bl)I+ble⊤,l=1,…,L, K_l=(1-b_l)I+b_le_ m1 , l=1,…,L, so the transition matrix from time 00 to tlt_l is K¯l=KlKl−1⋯K1=αlI+(1−αl)e⊤, K_l=K_lK_l-1·s K_1= _lI+(1- _l)e_ m1 , for αl=∏r=1l(1−br) _l= _r=1^l(1-b_r). The conditional one-step time reversal from tlt_l to tl−1t_l-1 has the closed form Kl←,x0 K_l^←,x_0 =I+(αl−1−αl)/(1−αl)⋅(ex0−e)e⊤,l=1,…,L, =I+( _l-1- _l)/(1- _l)·(e_x_0-e_ m)e_ m , l=1,…,L, with kernel qtl,tl−1(y∣x,x0)=ℙ(Xtl−1=y∣Xtl=x,X0=x0)=ey⊤Kl←,x0exq_t_l,t_l-1(y x,x_0)=P(X_t_l-1=y X_t_l=x,X_0=x_0)=e_y K_l^←,x_0e_x. Equivalently, qtl,tl−1(y∣x,x0) q_t_l,t_l-1(y x,x_0) =(αl−1−αl)/(1−αl),x=,y=x0,(1−αl−1)/(1−αl),x=,y=,y=x,x≠, = cases( _l-1- _l)/(1- _l),&x= m,\ y=x_0,\\ (1- _l-1)/(1- _l),&x= m,\ y= m,\\ 1\y=x\,&x≠ m, cases so an already unmasked token remains fixed while a masked token is either kept masked or revealed as its original clean value. Conditioned on the clean sequence, the true one-step reverse kernel factorizes across coordinates, qtl,tl−1(tl−1∣tl,0)=∏j=1Nqtl,tl−1(xtl−1j∣xtlj,x0j)q_t_l,t_l-1(x_t_l-1 _t_l,x_0)= _j=1^Nq_t_l,t_l-1(x_t_l-1^j x_t_l^j,x_0^j), which is the analytic posterior that (2.1) approximates. The same derivation covers the reverse kernel used by the standard masked diffusion sampler in Appendix E. Appendix G Per-output NFE Breakdown We record the per-output number of function evaluations (NFE), where one NFE counts one sequence-level masked discrete diffusion forward pass, not one vectorized invocation on a batch. This remains a model-only accounting convention: reward evaluations on completed sequences are reported separately and are not included in NFE. Batching can combine many sequence-level evaluations into one physical call without reducing this count; actual latency also depends on hardware, batch size, memory use, and reward-model latency. Let |⋅||·| denote set cardinality, ssplits_split the number of reverse updates before the best-of-K split, and [lend,l∗][l_end,l^*] the guidance window from Appendix H. For update index s∈0,…,L−1s∈\0,…,L-1\, the corresponding reverse state index is l=L−sl=L-s. Define =s:lend≤L−s≤l∗G=\s:l_end≤ L-s≤ l^*\ as the set of guided updates, with s∈0,…,L−1s∈\0,…,L-1\ throughout. Let the active branch multiplicity be bs=1b_s=1 for s<ssplits<s_split and bs=Kb_s=K for s≥ssplits≥ s_split, and set S=∑s∈bsS= _s b_s. Under the stated one-head-pass-per-candidate implementation, the per-output model NFE is: NFE =|s:s<ssplit|+K⋅|s:ssplit≤s<L| =|\s:s<s_split\|+K·|\s:s_split≤ s<L\| +M⋅|s:s∈,s<ssplit|+K⋅M⋅|s:s∈,ssplit≤s<L|. +M·|\s:s ,\,s<s_split\|+K· M·|\s:s ,\,s_split≤ s<L\|. (G.1) The first line counts the shared base masked discrete diffusion forward pass on ¯tl x_t_l at every update (one pass per update before the split, K passes per update after). The second line counts the M candidate prediction-head passes for h-estimation inside the guidance window, also with the K×K× multiplier after the split. Equivalently, NFE=∑s=0L−1bs+MSNFE= _s=0^L-1b_s+MS. Drawing nhn_h completions per candidate leaves this model NFE unchanged when all completions reuse that candidate’s prediction-head probabilities; the executed single-completion sampler uses nh=1n_h=1. The corresponding number of scored candidate completions per output is candidate reward evaluations=nhMS. reward evaluations=n_hMS. If K>1K>1 and a split occurs, terminal ranking requires at most K additional reward evaluations per output when scores of the final sequences are not cached; candidate-completion scores can be reused only when they score those same final sequences. The number of physical batched reward calls is implementation-dependent and is not this sequence-level evaluation count. Setting K=1K=1 or ssplit=Ls_split=L disables the split and its terminal ranking, recovering the core D-DOIT branch. These counts describe computational work rather than a wall-clock speed guarantee. Appendix H Detailed Sampler Pseudocode Appendix H gives the outer sampling loop and Appendix H the per-step guided update. Algorithm 2 D-DOIT with optional best-of-K refinement. The guided step is defined in Appendix H; setting K=1K=1 or ssplit=Ls_split=L disables the split. Copies share all tokens revealed before the split; final reward selection changes the output law. 0: Frozen PθP_θ, reward R, batch size B, steps L, guidance window [lend,l∗][l_end,l^*], split-update index ssplits_split, width K, guided-step parameters (M,ω,β,τ,γ)(M,ω,β,τ,γ) 1: Initialize B all- m trajectories at time tLt_L. 2: for s=0,1,…,L−1s=0,1,…,L-1 do 3: If s=ssplits=s_split and K>1K>1, duplicate each active trajectory into K copies. 4: Set the reverse state index l=L−sl=L-s. 5: Apply DoobGuidedStep to active trajectories when lend≤l≤l∗l_end≤ l≤ l^*; otherwise apply the standard transition (2.1). 6: end for 7: If K>1K>1 and ssplit<Ls_split<L, select the highest-reward clean copy in each group by (3.9); otherwise return the generated clean sequences. Algorithm 3 DoobGuidedStep: the practical single-completion update, without proposal-ratio correction. 0: current state ¯tl x_t_l, step index l, frozen model θ, reward R; (M,ω,β,τ,γ)(M,ω,β,τ,γ) 1: Compute αl _l and αl−1 _l-1 from the noise schedule schedule constants 2: Proposal pass: compute μθj(⋅∣¯tl,tl) _θ^j(· x_t_l,t_l) for all j 3: Form the τ-tempered proposal head μθ,τj(v∣¯tl,tl)∝μθj(v∣¯tl,tl)1/τ _θ,τ^j(v x_t_l,t_l) _θ^j(v x_t_l,t_l)^1/τ 4: Draw ¯tl−1(m)m=1M\ y_t_l-1^(m)\_m=1^M from (2.1) with μθ _θ replaced by μθ,τ _θ,τ candidate reverse states 5: for m=1,…,Mm=1,…,M do 6: Draw ^0(m)∼pθ,γhead(⋅∣¯tl−1(m)) x_0^(m) p^head_θ,γ(· y_t_l-1^(m)) via (3.3) prediction-head completion 7: h^(¯tl−1(m),tl−1)←exp(R(^0(m))/β) h( y_t_l-1^(m),t_l-1)← (R( x_0^(m))/β ) random head-proxy value 8: end for 9: Select m∗m^* by the weights in (3.7); return ¯tl−1=¯tl−1(m∗) x_t_l-1= y_t_l-1^(m^*) Appendix I Additional Analyses The main experiments evaluate HepG2 enhancer design, cell-type-specific regulatory DNA design, and protein inverse folding. This appendix collects approximation diagnostics and oracle-transfer analyses; the compute–quality trade-off is discussed in the main text. Matched-time sampling controls are detailed in Sections J.3 and K.5; full diagnostic sweeps appear in Appendix L. Concentrated but stochastic selection. At M=8M=8 and ω=8ω=8, mean ESS is 1.143±0.0021.143± 0.002 and the top-weighted candidate is selected on 95.72%95.72\% of guided steps. Soft resampling reaches final guidance reward 5.237±0.0515.237± 0.051, versus 5.154±0.0275.154± 0.027 for direct argmax in the same diagnostic (Section L.6). Thus selection is near-greedy but not deterministic; low ESS measures local weight concentration, not final-sequence diversity. Suffix search and nonmonotone progress. With M=9M=9, K=4K=4, and nh=1n_h=1 fixed, moving the split from update 105105 to 00 increases final guidance reward from 5.272±0.0575.272± 0.057 to 5.890±0.0285.890± 0.028, but raises wall time from 326.4±0.5326.4± 0.5 to 793.6±2.2793.6± 2.2 seconds per job. Even with 90.38%90.38\% of bases fixed, independent suffix completions differ at 66.1%66.1\% of the remaining positions. Late branching therefore searches a diverse suffix conditional on a fixed prefix; it cannot revise pre-split reveals. Separately, held-out activity of argmax-completed partial states is nonmonotone despite higher final activity than frozen sampling (Section L.7). These completion-proxy scores are distinct from final guidance reward R and the batch-median Pred-Activity benchmark. The diagnostics characterize practical search trade-offs, not exact reward-tilted sampling. Appendix J Experimental Details of Regulatory DNA Sequence Design In this section, we provide the experimental details and additional experimental analysis for the regulatory DNA sequence design task in Section 4.1. J.1 Dataset and Training Details Dataset. We conduct experiments on the HepG2 enhancer corpus from Gosai et al. [24]. The corpus contains ∼700 700k 200200 bp human enhancer sequences with measured HepG2 activity. We follow the dataset split of Wang et al. [8] so that our evaluation seeds match the DRAKES test partition and our numbers compare against the published DRAKES table. Reward oracle setup. We use two Enformer-based oracles [64] from Wang et al. [8]. The two oracles use disjoint chromosome splits of the same corpus. The finetuning oracle provides the reward signal R(0)R(x_0) during guided sampling. The held-out evaluation oracle scores Pred-Activity at test time. This matches the DRAKES protocol and makes our numbers comparable to the TDS, SMC, CG, and DRAKES results in Wang et al. [8]. Pre-training setup. D-DOIT operates on a frozen pre-trained masked discrete diffusion model [3]. We do not update its parameters during sampling. We use the public masked discrete diffusion enhancer checkpoint of Wang et al. [8], which the authors pre-train on the ∼700 700k enhancer corpus. Sampling setup. We use the two operating points defined in Section 3.2: D-DOIT-LBOK-S with a small per-step budget plus a late-stage trajectory split, and D-DOIT-Q with a larger inference-time budget for quality. Table 5 lists the exact hyperparameter values per benchmark. All main DRAKES rows use a single A100 80GB GPU and follow the DRAKES evaluation protocol with 640640 sequences per seed. Most rows use 33 random seeds; the re-measured TDS and the adopted D-DOIT-LBOK-S rows use 55 fresh final-test seeds after held-out eval selection. The TDS and D-DOIT-LBOK-S rows are re-measured on an A100-SXM4-80GB GPU; the remaining rows are reported as A100 reruns. Seed identity alone does not guarantee bitwise agreement across hardware or software implementations, so comparisons also require the same checkpoint, sampler, and evaluation protocol. We do not copy numbers from [8], so wall-clock and reward differences against the published table reflect our hardware and seed protocol rather than reporting drift. Table 5: Hyperparameters of D-DOIT. We report two operating points per benchmark: Q = quality-oriented and S = speed-oriented. Q denotes high-budget quality; S denotes LBOK-S. Selection and test protocols are described in Tables 2 and 3; diagnostic sweeps specify their own configurations. DRAKES Ctrl-DNA Sym. Name Role Range Q S Q S L Steps Total discrete reverse steps. 32,64,128\32,64,128\ 128128 128128 128128 128128 M Budget Candidate next states per guided step; linear head-evaluation cost. 2,…,64\2,…,64\ 5252 99 5252 99 ω Strength Exponent on h h in (3.7); for one completion, log-weight sharpness is ω/βω/β. [0.5, 8.0][0.5,\,8.0] 3.03.0 3.03.0 8.08.0 3.03.0 β Temp. Scales reward: h^=exp(R/β) h= (R/β) in (3.3). Small β amplifies differences. [0.1, 2.0][0.1,\,2.0] 0.20.2 0.20.2 0.50.5 0.20.2 τ Prop. T Proposal temperature (3.6) for candidate diversity; changes the reference kernel. [1.0, 3.0][1.0,\,3.0] 3.03.0 3.03.0 3.03.0 3.03.0 γ Comp. T Completion temperature for prediction-head completion ^0 x_0 in (3.3); γ<1γ<1 sharpens signal. [0.5, 2.0][0.5,\,2.0] 1.21.2 1.21.2 1.21.2 1.21.2 l∗l^* Guided high Highest reverse index where guidance is active; guided steps satisfy lend≤l≤l∗l_end≤ l≤ l^*. L/4,…,L\L/4,…,L\ L L L L lendl_end Guided low Lowest reverse index where guidance remains active; guided steps satisfy lend≤l≤l∗l_end≤ l≤ l^*. 1,4,8\1,4,8\ 11 11 11 11 ssplits_split Split idx. Number of reverse updates before each trajectory splits into K copies; 00 applies best-of-K from the start. 0,0.5L,…,L\0,0.5L,…,L\ 00 105105 6464 105105 K Width Number of copies per split trajectory (3.9). 1,2,4,8,20\1,2,4,8,20\ 88 44 2020 44 B Batch Physical output batch; affects memory use and measured throughput. — 3232 6464 256256 256256 nhn_h Completions Independent head completions per candidate; distinct from sequence length N. 1,2,3\1,2,3\ 11 11 11 11 J.2 Evaluation Metrics and Baselines Evaluation metrics. We use four metrics that cover predicted activity, chromatin accessibility, and sequence naturalness: • Predicted activity (Pred-Activity): We use the held-out evaluation reward oracle of [8] to estimate HepG2 enhancer activity for each generated sequence and report the median across the batch following the DRAKES convention. Higher scores indicate more functional sequences. • Chromatin accessibility (ATAC-Acc): We use an independent binary classifier trained on ENCODE HepG2 chromatin accessibility data [65] to test whether the designed sequences fall in accessible chromatin regions. This is a key property of active enhancers. Higher accuracy indicates more active sequences. • 3-mer pearson correlation (3-mer Corr): We compute the Pearson correlation between the 33-mer frequency profile of the generated sequences and that of natural HepG2 enhancers from [24]. Higher correlation indicates that the generated sequences match the overall distribution of natural enhancer sequences. • JASPAR motif correlation (JASPAR Corr): We compute the Spearman correlation between the transcription-factor binding motif frequencies of the generated sequences and the natural enhancer reference, scored against the JASPAR motif database [66]. Higher correlation indicates better preservation of regulatory motif structure. Baselines. We compare D-DOIT against training-free and training-based methods. Training-free methods steer a frozen pre-trained model at inference time: • Pre-trained: the unguided masked discrete diffusion model of [8]; provides the unguided reward and naturalness reference. • CG [10]: classifier guidance with discrete gradients; backpropagates through a Gumbel-Softmax relaxation and requires a differentiable reward. • SMC and TDS [11]: particle-based guidance with path likelihood ratios. TDS is a strong reported training-free baseline and provides the runtime reference for LBOK-S; its budget is not matched to Q. • SVDD-PM [12]: posterior-mean value decoding, using the same frozen MDLM and oracle protocol, with M=416M=416 for similar runtime to Q. • Frozen Best-of-NBoNN_BoN: endpoint-only selection over independent unguided samples; NBoN=732N_BoN=732 in the Q wall-clock comparison. Training-based methods update model parameters: • DRAKES [8]: reward finetunes the pre-trained masked discrete diffusion model with a KL regularizer; we treat it as the strongest naturalness-preserving training-based reference. • DRAKES w/o KL: removes the KL term to expose the over-optimization risk of unregularized reward maximization. • CFG [67]: trains a conditional model from scratch. J.3 Nearest-neighbor and Endpoint-search Controls The SVDD-PM comparison uses the same frozen MDLM, guidance Enformer, held-out evaluator, paired seeds 100100–102102, and 640640 final outputs per method and seed. We calibrate its candidate count to M=416M=416 to give a similar wall-clock budget to Q (L=128,M=52,K=8,ssplit=0L=128,M=52,K=8,s_split=0). Pred-Activity is computed as a median within each seed, then summarized by the mean and sample standard deviation across seeds. The measured times are 10.59510.595 and 11.23611.236 seconds per output for Q and SVDD-PM, respectively. Q has higher means on all four metrics; the clearest separation is ATAC-Acc (91.67%91.67\% versus 39.84%39.84\%), while Pred-Activity differs by 0.150.15. Table 6: Matched-time controls. We test D-DOIT-Q vs SVDD-PM and Frozen Best-of-732 at similar measured wall-clock cost. Bold marks the best mean per column. Metric D-DOIT-Q SVDD-PM BoN s/output ↓ 10.59510.595 11.23611.236 10.58710.587 Activity ↑ 6.19±0.056.19± 0.05 6.04±0.076.04± 0.07 4.70±0.034.70± 0.03 ATAC (%) ↑ 91.7±0.991.7± 0.9 39.8±0.439.8± 0.4 43.5±1.343.5± 1.3 3-mer ↑ 0.894±0.0090.894± 0.009 0.688±0.0120.688± 0.012 0.881±0.0080.881± 0.008 JASPAR ↑ 0.923±0.0020.923± 0.002 0.917±0.0020.917± 0.002 0.957±0.0020.957± 0.002 Frozen Best-of-NBoNN_BoN samples independent complete sequences from the same unguided backbone, scores them with the guidance Enformer, and retains the highest-scoring endpoint in each group. A separate held-out Enformer evaluates the retained sequences. Same-A100 calibration gives NBoN=732N_BoN=732, with 10.58710.587 seconds per output compared with Q’s 10.59510.595. Table 6 reports this high-budget comparison independently of the lower-budget diagnostics; Q improves activity, accessibility, and 3-mer correlation, while endpoint selection achieves higher JASPAR correlation (0.9570.957 versus 0.9230.923). This control tests whether spending the same time on more complete unguided endpoints can replace local value-guided transitions. Endpoint selection does not reproduce the activity and accessibility gains, although it better preserves this motif statistic. Reward evaluation is included in the measured wall-clock cost. Separating model and reward budgets. Table 7 additionally compares diagnostic settings with five seeds and 640640 outputs per seed. The branched M=9M=9 setting uses 1,9701,970 diffusion evaluations per output, close to Best-of-15’s 1,9201,920, but incurs substantially more oracle work. Its 1,7771,777 reward evaluations include 1,7731,773 candidate completions and four terminal scores. Across a 640640-output job, batching combines 1,137,2801,137,280 scored sequences into 1,2901,290 vectorized oracle calls. These counts explain why matching only backbone evaluations does not match wall-clock time. Table 7: Backbone work, oracle work, and measured time on DRAKES. Each setting uses five sampling seeds and 640640 outputs per seed. Pred-Activity is held-out Enformer activity; entries report means and sample standard deviations across seeds. The final column measures the full 640640-output job and must not be extrapolated to other physical batch sizes. The unbranched and branched rows are diagnostic settings, not substitutions for the separately selected main-table LBOK-S or Q results. Configuration Pred-Activity ↑ Model NFE per output Reward evaluations per output s/640 outputs Frozen Best-of-15 2.219±0.0172.219± 0.017 1,9201,920 1515 97.8±1.597.8± 1.5 M=9,τ=1,K=1M=9,τ=1,K=1 3.115±0.0213.115± 0.021 1,2801,280 1,1521,152 227.0±1.9227.0± 1.9 M=9,τ=3,K=1M=9,τ=3,K=1 3.670±0.0723.670± 0.072 1,2801,280 1,1521,152 227.8±2.6227.8± 2.6 M=9,τ=3,K=4,ssplit=105M=9,τ=3,K=4,s_split=105 4.491±0.0454.491± 0.045 1,9701,970 1,7771,777 327.8±4.3327.8± 4.3 J.4 Guidance-oracle and Evaluator Robustness We distinguish two controls, both using five sampling seeds and 640640 sequences per seed. In Table 8, only the guidance oracle changes among Enformer, CNN, and MLP; sampler settings, seeds, output count, held-out Enformer, and biological metrics remain fixed. All three guidance oracles improve held-out activity and accessibility over the frozen model, but the effect size varies with the oracle. Enformer guidance gives the highest activity and accessibility, whereas MLP guidance gives the highest 3-mer correlation in this diagnostic. In Table 9, we instead rescore the same Enformer-guided and unguided sequences with held-out Enformer, CNN, and MLP predictors. D-DOIT exceeds the frozen model within each evaluator. Because their output scales differ, cross-column score magnitudes are not directly comparable. These controls assess predictor transfer separately from the selected high-budget Q row and do not substitute for experimental biological assays. Table 8: Changing the guidance oracle. Only the guidance reward R changes among the D-DOIT rows; the sampler, seeds, output count, held-out Enformer evaluator, and other metrics are fixed. Entries are mean ± sample standard deviation over five seeds, with 640640 sequences per seed. Pred-Activity uses the held-out Enformer. These results are separate from the high-budget Q evaluation. Generation method Pred-Activity ↑ ATAC-Acc (%) ↑ 3-mer Corr ↑ JASPAR Corr ↑ Enformer-guided D-DOIT 4.491±0.0454.491± 0.045 68.25±1.8668.25± 1.86 0.8630±0.01600.8630± 0.0160 0.9468±0.00250.9468± 0.0025 CNN-guided D-DOIT 2.203±0.0542.203± 0.054 51.91±1.1951.91± 1.19 0.8738±0.00130.8738± 0.0013 0.9342±0.00210.9342± 0.0021 MLP-guided D-DOIT 1.544±0.0371.544± 0.037 28.78±1.2928.78± 1.29 0.9489±0.00350.9489± 0.0035 0.9347±0.00310.9347± 0.0031 Frozen pre-trained model 0.371±0.0180.371± 0.018 1.66±0.591.66± 0.59 −0.0584±0.0149-0.0584± 0.0149 0.2723±0.02200.2723± 0.0220 Table 9: Transfer across held-out evaluators. The same Enformer-guided D-DOIT and frozen-model sequences are rescored by three held-out predictors. Entries are mean ± sample standard deviation over five seeds, with 640640 sequences per seed. Enformer scores are Pred-Activity; CNN and MLP scores have their own scales. Compare methods within each column, not absolute values across columns. Generation method Pred-Activity ↑ (held-out Enformer) Held-out CNN ↑ Held-out MLP ↑ Enformer-guided D-DOIT 4.491±0.0454.491± 0.045 0.867±0.0140.867± 0.014 0.634±0.0150.634± 0.015 Frozen pre-trained model 0.371±0.0180.371± 0.018 0.484±0.0140.484± 0.014 0.509±0.0140.509± 0.014 J.5 Additional Experimental Analysis For the experimental results in Table 2, we provide the additional detailed analysis below. Section 4.1 reports the headline TDS comparisons. We add two pieces of detail here. First, the D-DOIT-LBOK-S vs. TDS gap on ATAC-Acc is +25.0+25.0 percentage points (40.0%→65.0%40.0\%→ 65.0\%), while Pred-Activity is matched (4.564.56 vs. 4.584.58) with lower reported seed variability; this is descriptive, not a variance-test result. The same operating point also lifts ATAC-Acc from 39.9%39.9\% to 65.0%65.0\% over SMC and JASPAR Corr from 0.7560.756 to 0.9480.948 over SMC. Second, high-budget D-DOIT-Q over TDS gives +1.61+1.61 in Pred-Activity, +51.7+51.7 percentage points in ATAC-Acc, +0.077+0.077 in 3-mer Corr, and +0.124+0.124 in JASPAR Corr. Compared with the training-based DRAKES, D-DOIT-Q achieves higher Pred-Activity (6.196.19 vs. 5.615.61), higher 3-mer Corr (0.8940.894 vs. 0.8870.887), and higher JASPAR Corr (0.9230.923 vs. 0.9110.911) without any backbone update. DRAKES remains higher on ATAC-Acc (92.5%92.5\% vs. 91.7%91.7\%), so we do not claim an all-metric win. The unregularized variant DRAKES w/o KL exposes the cost of training-based reward maximization: it gains +0.83+0.83 in Pred-Activity over DRAKES but loses 0.5800.580 in 3-mer Corr (0.887→0.3070.887→ 0.307) and 0.3540.354 in JASPAR Corr (0.911→0.5570.911→ 0.557). D-DOIT-Q stays near the DRAKES high-naturalness regime. The frozen model supplies candidate proposals without retraining. The exact reference transform reweights its induced distribution; the practical sampler’s temperatures and selection operations change that reference law. Figure 2 plots utility and naturalness for the original benchmark comparison set. The separately time-calibrated SVDD and endpoint controls are reported in Table 6, rather than pooled into a claim of an exhaustive compute–quality frontier. D-DOIT-LBOK-S anchors the practical moderate-budget side of this frontier. D-DOIT-Q occupies the high-utility, high-naturalness corner when inference-time budget is deliberately relaxed, with ATAC-Acc as the remaining gap to DRAKES. J.6 Held-out Final-test Check for D-DOIT-LBOK-S We select the DRAKES LBOK-S configuration on eval seeds 42,43,4442,43,44 and report it once on fresh final-test seeds 105,106,107,108,109105,106,107,108,109. The fixed final configuration uses M=9M=9, K=4K=4, ssplit=105s_split=105, ω=3.0ω=3.0, β=0.2β=0.2, τ=3.0τ=3.0, and γ=1.2γ=1.2. It exceeds the TDS thresholds on ATAC-Acc, 3-mer Corr, and JASPAR Corr, and has a similar mean Pred-Activity to TDS; no statistical equivalence test is implied. Table 10: Held-out final-test check for D-DOIT-LBOK-S. Targets are the TDS row in Table 2. The labels compare reported means with the reference values; they are not significance or statistical-equivalence decisions. Metric TDS reference D-DOIT-LBOK-S final test Mean comparison Pred-Activity 4.584.58 4.560(0.040)4.560(0.040) similar mean ATAC-Acc 40.0%40.0\% 65.03(1.43)%65.03(1.43)\% pass 3-mer Corr 0.8170.817 0.8511(0.0165)0.8511(0.0165) pass JASPAR Corr 0.7980.798 0.9477(0.0016)0.9477(0.0016) pass J.7 Held-out Selection for High-budget D-DOIT-Q To avoid tuning directly on the final reported seeds, we separate the DRAKES high-budget update into three seed groups. Development seeds 0,1,20,1,2 identify promising sampler families only. Held-out eval seeds 42,43,4442,43,44 select the final high-budget D-DOIT-Q configuration (M=52M=52, K=8K=8, ssplit=0s_split=0, ω=3.0ω=3.0, β=0.2β=0.2, τ=3.0τ=3.0, γ=1.2γ=1.2). Final-test seeds 100,101,102100,101,102 are then evaluated once and reported as the held-out final numbers in Table 2. Table 11: Held-out final-test check for D-DOIT-Q. Targets are the DRAKES row in Table 2. The final-test result passes Pred-Activity, 3-mer Corr, and JASPAR Corr, but misses the ATAC-Acc target on the held-out evaluation. Metric DRAKES reference D-DOIT-Q final test Mean comparison Pred-Activity 5.615.61 6.19(0.05)6.19(0.05) pass ATAC-Acc 92.5%92.5\% 91.7(0.9)%91.7(0.9)\% miss 3-mer Corr 0.8870.887 0.894(0.009)0.894(0.009) pass JASPAR Corr 0.9110.911 0.923(0.002)0.923(0.002) pass Figure 2: Utility-naturalness trade-off. We plot Pred-Activity (median) against 3-mer Corr; one point per method. Green shading marks the high-utility, high-naturalness region. This view covers the original benchmark rows; the separately time-calibrated SVDD and endpoint comparisons are in Table 6. ATAC-Acc is reported separately in Tables 2 and 11. Appendix K Experimental Details of Cell-type-specific Enhancer Design In this section, we provide the experimental details and additional experimental analysis for the cell-type-specific enhancer design task in Section 4.2. K.1 Dataset and Training Details Dataset. We use the 200200 bp human enhancer corpus of Chen et al. [25], covering three cell lines: HepG2, K562, and SK-N-SH. Data preprocessing follows the regLM protocol of Lal et al. [26], as in Chen et al. [25]. We use the Ctrl-DNA enhancer task only for its data, cell-type-specific reward oracles, and evaluation metrics. The training procedure and backbone architecture come from our own setup, which we describe in the next paragraphs. Reward oracle setup. We use the cell-type-specific Enformer-based oracles [64] from Chen et al. [25], which the authors train on the same corpus. Unlike constrained-RL baselines with explicit thresholds and policy retraining, D-DOIT is an inference-time method and accepts a single black-box scalar reward. For target cell line c and off-target cell lines −cC_-c, we set R(0)=Rc(0)−λ⋅maxc′∈−cRc′(0), R(x_0)=R_c(x_0)-λ· _c _-cR_c (x_0), (K.1) where each RcR_c is the cell-type-specific Enformer oracle of Chen et al. [25]. The max over off-targets suppresses the worst case during sampling. The evaluation ΔR R averages over off-targets to follow the Ctrl-DNA convention. We sweep λ on held-out eval seeds disjoint from the final-test seeds. The final high-budget row uses the task-default penalties selected for the Q operating point, and the final low-budget LBOK-S row uses λ=2.0λ=2.0 selected by the low-budget eval sweep. Pre-training setup. We train a new masked discrete diffusion model [3] on the Ctrl-DNA enhancer corpus. The HyenaDNA backbone used by Chen et al. [25] is autoregressive and does not fit our discrete h-transform. D-DOIT operates on this frozen model without updating its parameters during sampling. The same checkpoint serves all three cell lines and both operating points. Sampling setup. We use the two operating points from Section 3.2. Table 5 lists the exact hyperparameter values per benchmark. For the D-DOIT Ctrl-DNA rows, we choose fixed configs on eval seeds 42,43,4442,43,44 and report final-test seeds 100,101,102,103,104100,101,102,103,104. We use 55 random seeds with batch size 256256 per seed and run a separate set for each of the three target cell lines. Pre-trained, CG, SMC, and TDS are the rerun training-free comparison rows. Other baselines come from the reported results in Chen et al. [25] Table 1 under the 55-seed, δ=0.5δ=0.5 setting. K.2 Evaluation Metrics and Baselines Evaluation metrics. We use three metrics that cover cell-type specificity, motif structure, and set-level diversity: • Cell-specific reward difference (ΔR R): We compute the difference between the target-cell oracle score and the average of the off-target oracle scores following the Ctrl-DNA convention. Higher scores indicate stronger cell-type specificity. • Motif correlation (Motif Corr): We compute the Pearson correlation between the transcription-factor binding-site frequencies of the generated sequences and the cell-type-specific natural enhancer set from [25] (top half by target activity, bottom half by off-target activity). Higher correlation indicates that the generated sequences match the regulatory grammar of natural cell-specific enhancers. • Set-level diversity (Diversity): We compute the Shannon entropy across the generated sequence set with the official Ctrl-DNA evaluation script [25]. Higher entropy indicates that the sampler does not collapse onto a small set of high-reward sequences. ΔR R and Motif Corr appear in Table 3. Diversity appears in Figure 3. We omit Median because [25] mentions it in text but does not report numerical values. Baselines. We compare D-DOIT against the baselines from Chen et al. [25] and the same training-free guidance methods as in Section J.2: • Pre-trained, CG [10], SMC, TDS [11]: training-free inference-time methods on the same frozen masked discrete diffusion model. These share the frozen backbone with D-DOIT. Frozen Best-of-708 additionally provides the same-A100 endpoint-search control at Q’s measured time budget. • regLM [26]: supervised generative baseline that the authors train on the cell-type-specific enhancer corpus. • PPO, TACO, PPO-Lagrangian [25]: RL-based methods that retrain the generator against the specificity reward. • AdaLead, PEX, BO, CMA-ES [25]: directed-evolution methods that search the sequence space directly without any generative backbone. K.3 Training-free Baseline Tuning Screen To check that the Ctrl-DNA training-free baselines are not weak only because of arbitrary default hyperparameters, we run a bounded seed-42 screen for CG, SMC, and TDS on HepG2, K562, and SK-N-SH. Table 12 summarizes all 13 screened configurations at the family level and names the selected follow-up configuration for each baseline family. Across this screen, CG remains a high-motif, near-zero-reward baseline; SMC obtains only small reward movement and degrades motif/diversity; and TDS remains weak on the composite specificity reward. Because this is a one-seed screen rather than a final-test protocol, we use it as baseline-tuning provenance instead of replacing the main-table baseline rows reported above. Table 12: Ctrl-DNA training-free baseline tuning screen. We run a bounded seed-42 screen over 13 CG, SMC, and TDS hyperparameter settings on all three Ctrl-DNA cell tasks. The screen documents that these training-free baselines remain weak after tuning: CG preserves motifs without improving reward/specificity, SMC gains little reward while degrading motif/diversity, and TDS stays near zero reward movement. The selected configs are provenance for possible follow-up evaluation, not final-test rows. Method Seed-42 screen Selected config Screen outcome Takeaway CG guidance scale 20,50,100,200\20,50,100,200\ CG_G20: guidance scale 2020 ΔR∈[−0.002,0.000] R∈[-0.002,0.000]; specificity ≤−0.007≤-0.007; motif corr 0.9030.903–0.9160.916 All screened CG settings preserve motifs but do not improve reward or specificity. SMC beta values: 0.30.3, 0.50.5, 0.750.75, 1.01.0, 1.51.5 SMC_B05: beta 0.50.5 Best ΔR=0.011 R=0.011 at beta 1.51.5, with motif corr 0.3880.388 and diversity 0.9280.928; selected beta 0.50.5 has ΔR=0.006 R=0.006. Small reward gains come with much weaker motif/diversity, so SMC is not competitive. TDS g/beta pairs: 2/0.752/0.75, 5/0.755/0.75, 5/1.05/1.0, 10/1.010/1.0 TDS_G5_B10: g=5g=5, beta 1.01.0 ΔR∈[−0.002,0.001] R∈[-0.002,0.001]; specificity <0<0 for all settings; best motif corr 0.5100.510. TDS remains weak after this bounded tuning pass. K.4 Additional Experimental Analysis For the experimental results in Table 3, we provide the additional detailed analysis below. Table 3 records positive mean specificity for the two D-DOIT configurations relative to the originally reported CG, SMC, and TDS rows. The time-calibrated endpoint control additionally attains positive specificity, with mean ΔR=0.108 R=0.108. Compared with TDS, D-DOIT-LBOK-S improves ΔR R from 0.0020.002 to 0.0330.033 and improves Motif Corr from 0.5730.573 to 0.6480.648, at about 0.94×0.94× the rerun TDS wall-clock in the Ctrl-DNA timing logs. CG fails to move the specificity reward at all (ΔR≈0 R≈ 0, indistinguishable from Pre-trained), and SMC sits at ΔR=0.000 R=0.000 with Motif Corr 0.5870.587. Q reaches ΔR=0.119±0.063 R=0.119± 0.063 and Motif Corr 0.498±0.0700.498± 0.070. Its 5.435.43 seconds per output is close to Frozen Best-of-708’s 5.515.51 on the same A100. The small mean advantage over endpoint selection is not claimed statistically significant. Table 13 provides the per-cell breakdown for the original guidance comparisons. Compared with the remaining training-based baselines, D-DOIT-Q stays behind regLM on ΔR R (0.1190.119 vs. 0.160.16). Because regLM uses a different generative backbone and training procedure, this gap does not isolate the effect of inference-time adaptation. On Motif Corr, D-DOIT-Q at 0.4980.498 exceeds every remaining training-based and evolutionary baseline (PPO-Lagrangian 0.420.42, AdaLead 0.360.36). These means describe motif preservation at the reported specificity levels; they do not identify which mechanism caused the difference. Pre-trained has high Motif Corr (0.9370.937) at ΔR≈0 R≈ 0, providing an unguided naturalness reference rather than a mathematical ceiling. LBOK-S has higher Motif Corr (0.6480.648) than Q and Frozen Best-of-708, while achieving lower specificity. D-DOIT-Q (0.4980.498) trades motif fidelity for higher ΔR R, so we use it as the high-specificity operating point rather than a motif-preserving one. Figure 3 compares diversity and specificity within the original guidance baseline set; the endpoint control’s reward and motif scores are compared separately in Table 3. SMC and TDS gain little ΔR R and collapse Shannon entropy to ≈1.5≈ 1.5. D-DOIT-Q occupies the high-ΔR R, high-diversity region across all three cell lines. Table 13 reports the per-cell breakdown of ΔR R, Motif Corr, and Diversity. Figure 3: Specificity vs. diversity on Ctrl-DNA. Each panel is one target cell line. Small markers are single seeds, and large black-edged markers are the corresponding 55-seed means. The x-axis is cell-specific reward gain ΔR R; the y-axis is set-level Shannon entropy from the Ctrl-DNA evaluation script [25]. The green region marks positive ΔR R with high diversity. D-DOIT moves into this region, whereas Pre-trained/CG remain near zero reward movement and SMC/TDS lose diversity. The headline averages in Table 3 are the cross-cell means of these rows. Both D-DOIT operating points achieve their largest ΔR R on K562, consistent with K562 carrying the largest reward headroom in the Ctrl-DNA setup, with smaller positive gains on HepG2 and SK-N-SH. D-DOIT-LBOK-S wins on Motif Corr for every cell line among training-free methods that move the reward. Both D-DOIT operating points keep Diversity high (≈1.98≈ 1.98) on all three cells. SMC and TDS drop to ≈1.5≈ 1.5. Table 13: Per-cell-type Ctrl-DNA results. We break down ΔR R, Motif Corr, and Diversity by target cell line. Parentheses denote standard deviation across 55 random seeds. Bold = best, underline = second best, per column within each cell line. Cell line Method ΔR R ↑ Motif Corr ↑ Diversity ↑ HepG2 Pre-trained −0.004(0.002)-0.004(0.002) 0.937(0.009)¯ 0.937(0.009) 1.988(0.001)1.988(0.001) CG −0.004(0.002)-0.004(0.002) 0.938(0.009)0.938(0.009) 1.988(0.001)1.988(0.001) SMC +0.003(0.011)+0.003(0.011) 0.582(0.094)0.582(0.094) 1.484(0.109)1.484(0.109) TDS −0.004(0.009)-0.004(0.009) 0.606(0.087)0.606(0.087) 1.509(0.061)1.509(0.061) D-DOIT-LBOK-S (Ours) +0.014(0.003)¯ +0.014(0.003) 0.667(0.032)0.667(0.032) 1.981(0.002)1.981(0.002) D-DOIT-Q (Ours) +0.095(0.008)+0.095(0.008) 0.558(0.048)0.558(0.048) 1.982(0.001)¯ 1.982(0.001) K562 Pre-trained +0.003(0.003)+0.003(0.003) 0.937(0.009)¯ 0.937(0.009) 1.988(0.001)1.988(0.001) CG +0.002(0.003)+0.002(0.003) 0.938(0.009)0.938(0.009) 1.988(0.001)1.988(0.001) SMC −0.006(0.006)-0.006(0.006) 0.613(0.133)0.613(0.133) 1.527(0.095)1.527(0.095) TDS −0.000(0.006)-0.000(0.006) 0.538(0.072)0.538(0.072) 1.546(0.084)1.546(0.084) D-DOIT-LBOK-S (Ours) +0.068(0.010)¯ +0.068(0.010) 0.620(0.031)0.620(0.031) 1.982(0.001)¯ 1.982(0.001) D-DOIT-Q (Ours) +0.204(0.009)+0.204(0.009) 0.421(0.037)0.421(0.037) 1.982(0.001)¯ 1.982(0.001) SK-N-SH Pre-trained +0.001(0.001)+0.001(0.001) 0.937(0.009)¯ 0.937(0.009) 1.988(0.001)1.988(0.001) CG +0.001(0.001)+0.001(0.001) 0.938(0.010)0.938(0.010) 1.988(0.001)1.988(0.001) SMC +0.004(0.009)+0.004(0.009) 0.566(0.133)0.566(0.133) 1.501(0.170)1.501(0.170) TDS +0.010(0.006)+0.010(0.006) 0.573(0.075)0.573(0.075) 1.508(0.088)1.508(0.088) D-DOIT-LBOK-S (Ours) +0.018(0.002)¯ +0.018(0.002) 0.658(0.054)0.658(0.054) 1.982(0.001)1.982(0.001) D-DOIT-Q (Ours) +0.057(0.003)+0.057(0.003) 0.517(0.035)0.517(0.035) 1.983(0.001)¯ 1.983(0.001) The final Ctrl-DNA Q operating point uses M=52M=52, K=20K=20, and ssplit=64s_split=64, whereas DRAKES Q uses M=52M=52, K=8K=8, and ssplit=0s_split=0. A preceding Ctrl-DNA eval sweep also considered M=8,K=20,ssplit=64M=8,K=20,s_split=64, which obtained config-level ΔR=0.117 R=0.117, and a M=32,K=8,ssplit=0M=32,K=8,s_split=0 setting with ΔR=0.105 R=0.105 at roughly three times the runtime. These exploratory settings differ from the final Q comparison and vary several parameters together. They illustrate alternative budget allocations, not an isolated causal effect of branching. The endpoint control shows that final selection alone already obtains much of the mean specificity gain under the calibrated budget. Table 14: Evolutionary-search baselines on Ctrl-DNA. We report ΔR R and Motif Corr averaged across the three target cell lines for the four directed-evolution baselines reproduced from [25]. The training-free and training-based comparisons appear in Table 3. Method ΔR R ↑ Motif Corr ↑ AdaLead 0.05(0.05)0.05(0.05) 0.36(0.08)0.36(0.08) PEX 0.09(0.02)0.09(0.02) 0.23(0.04)0.23(0.04) BO 0.01(0.02)0.01(0.02) 0.11(0.11)0.11(0.11) CMA-ES 0.01(0.01)0.01(0.01) 0.08(0.05)0.08(0.05) K.5 Wall-clock-matched Endpoint Control Frozen Best-of-708 generates complete unguided sequences from the frozen backbone and retains the best reward-scored endpoint in each group. We calibrate its group size on the same A100 used for Q (L=128,M=52,K=20,ssplit=64L=128,M=52,K=20,s_split=64). At 5.515.51 versus 5.435.43 seconds per output, the methods give specificity 0.108±0.02810.108± 0.0281 versus 0.119±0.0630.119± 0.063, and Motif Corr 0.490±0.1650.490± 0.165 versus 0.498±0.0700.498± 0.070. The means favor Q slightly, but the differences are modest and are not interpreted as statistically significant. Training-based rows remain cross-backbone references, separate from this inference-time comparison. Appendix L Ablation Study We conduct ablation studies on the DRAKES HepG2 enhancer task to understand the behavior of D-DOIT. We examine the guidance window l∗l^*, guidance strength ω and effective sample size (ESS), proposal and completion temperatures τ,γτ,γ, and candidate budget M. We also study completion count, direct-versus-rollout value estimates, soft-versus-argmax selection, split position, and reward trajectories (Tables 16, 17, 18, 19, 20 and 21). The window, strength, temperature, and candidate-budget sweeps use a single held-out seed unless noted otherwise; the additional diagnostics state their sampling protocols separately. Final guidance reward R is scored by the sampling oracle and is distinct from held-out Pred-Activity. Completion-proxy trajectories use the held-out evaluator but average scores of completed partial states rather than the benchmark’s within-seed batch median. We re-run the best γ-sweep configuration across the 33 main evaluation seeds and report it in Table 15. These studies show how D-DOIT balances reward optimization, sequence naturalness, and computational cost. The candidate-budget sweep shows that increasing M mainly improves functional utility, while naturalness metrics remain comparatively stable. The table surfaces three consequential design choices: the guidance window l∗l^* exhibits a phase transition for naturalness; the completion temperature γ controls prediction-head completion quality; and the candidate budget M controls utility. Table 15: Headline ablation findings. Each row varies one hyperparameter around the DRAKES ablation reference setting. We report sweep-margin endpoints from a single-seed sweep; full sweeps appear in Appendix L. Axis Setting 3-mer Corr Pred-Activity / ATAC (%) l∗l^* 64→12864→ 128 −0.06→0.894-0.06→ 0.894 3.61→4.773.61→ 4.77 / 33.6→72.733.6→ 72.7 γ 1.2→0.61.2→ 0.6 0.846→0.9050.846→ 0.905 4.53→4.724.53→ 4.72 / 72.7→66.472.7→ 66.4 M 8→328→ 32 +9%+9\% +31%+31\% / +29+29 p Runtime analysis. Section 4.4 reports the runtime comparison and its measurement conditions. The runtime pattern follows the NFE accounting in Appendix G: larger M and earlier best-of-K splitting increase candidate-state evaluations and oracle-scored completions. Increasing nhn_h also increases completion scoring, as quantified in Table 16. LBOK-S is the lower-budget operating point, whereas Q deliberately trades greater recurring inference cost for quality; neither setting implies cost dominance over training-based methods. L.1 Effect of Guidance Window l∗l^* Figure 4: Guidance-window sweep. Expanding the guided window improves utility and sharply recovers naturalness between l∗=96l^*=96 and l∗=112l^*=112. We study the effect of the guidance window l∗l^*, the highest reverse-time index where Doob’s h-transform is active in the discrete sampler. A smaller l∗l^* saves (L−l∗)⋅M(L-l^*)· M neural-function evaluations per sequence but omits guidance at more heavily masked states, where limited revealed context may affect the prediction-head proxy. Figure 4 reports the four DRAKES metrics as we sweep l∗∈10,20,…,128l^*∈\10,20,…,128\. The results show a phase-transition-like behavior in naturalness. All configurations with l∗≤50l^*≤ 50 produce negative 3-mer Pearson correlations. A sharp jump occurs between l∗=96l^*=96 (3-mer =0.21=0.21) and l∗=112l^*=112 (0.650.65). At l∗=128l^*=128, 3-mer reaches 0.8940.894, a 14×14× improvement over l∗=64l^*=64. Utility metrics improve across the same range without a sharp transition. In this sweep, broad guidance windows are associated with better global sequence-composition statistics; this observation does not isolate a causal mechanism or establish a requirement for other rewards. Utility metrics, in contrast, recover from a narrower window. We use l∗=128l^*=128 as the default setting. L.2 Effect of Guidance Strength ω and ESS Collapse We study the effect of the guidance strength ω in (3.7). A larger ω sharpens the transformed transition toward high-reward candidates. A smaller ω preserves more of the pre-trained kernel. Figure 5: ESS diagnostic. ESS and collapse fraction across ω; weighted selection collapses to near-greedy at ω≥4ω≥ 4. Figure 5 reports the per-step Effective Sample Size (ESS) of the candidate weights and the fraction of steps where ESS collapses below M/4M/4. The results show that ESS collapses to near 11 at ω≥4ω≥ 4. 95%95\% of guided steps have ESS <M/4<M/4 at ω=4ω=4, and 100%100\% at ω≥16ω≥ 16. A direct mechanism for concentration is reward-range amplification: ω=4ω=4 amplifies a candidate reward range of ≈3≈ 3 nats to ≥12≥ 12 nats and saturates the softmax. High-quality results remain achievable despite this collapse when proposal-level diversity (via τ) still places a strong candidate in the pool. This motivates joint tuning of ω and τ, but does not imply that larger τ recovers importance-weight accuracy. The concentration depends on the reward distribution and ω/βω/β, so these diagnostic thresholds cannot be transferred unchanged to other configurations. L.3 Effect of Proposal and Completion Temperatures τ,γτ,γ We then study the joint effect of the proposal temperature τ in (3.6) and the completion temperature γ in (3.3). A larger τ flattens the proposal and produces more diverse candidates. A smaller γ sharpens the prediction-head completion toward the model mode. Figure 6 reports the four DRAKES metrics across the joint sweep at M=32M=32. The results show a clear separation of roles. Functional activity peaks at higher proposal diversity. Marginal analysis shows Pred-Activity improves from 3.893.89 at τ=2.5τ=2.5 to 4.074.07 at τ=5.0τ=5.0, and ATAC-Acc from 53.7%53.7\% to 69.4%69.4\%. Naturalness peaks at sharper prediction-head completions. Reducing γ from 1.21.2 to 0.60.6 improves averaged 3-mer Corr from 0.8460.846 to 0.9050.905. At the strongest γ-sweep configuration (M=32M=32, τ=3.0τ=3.0, γ=0.6γ=0.6), 3-mer Corr reaches 0.949±0.0060.949± 0.006 across 33 seeds, above DRAKES at 0.8870.887. This sweep identifies the naturalness-preserving region. The final held-out DRAKES Q row in Table 2 uses a larger eval-selected inference budget and trades some naturalness for much higher Pred-Activity and ATAC-Acc while remaining above the DRAKES 3-mer and JASPAR targets. Naturalness metrics saturate around τ=3.5τ=3.5–4.04.0, suggesting a trade-off between naturalness and accessibility at extreme proposal temperatures. We use τ=3.0τ=3.0 and γ=0.6γ=0.6 as the ablation default for naturalness-preserving sweeps. Figure 6: Interaction of proposal and completion temperatures. We sweep the proposal temperature τ and the completion temperature γ at M=32M=32. Naturalness peaks at sharper prediction-head completions (smaller γ), while functional activity peaks at higher proposal diversity (larger τ). The red star marks the ablation reference configuration used before the final held-out Q budget selection. L.4 Effect of Candidate Budget M We study the effect of the Monte Carlo candidate budget M. The candidate count controls per-step work and coverage of possible next states. This sweep measures quality changes, not kernel error or convergence to the original tilted distribution. Figure 7 reports the four DRAKES metrics across M∈8,16,32M∈\8,16,32\. Figure 7: Candidate-budget sweep. We vary the Monte Carlo candidate budget M on the DRAKES HepG2 task and report the same four metrics. Increasing M gives direct gains in functional utility, while naturalness changes much less across the sweep. The results show that fewer candidates reduce functional utility but leave sequence naturalness almost unchanged. Reducing M from 3232 to 88 drops Pred-Activity by 31%31\% and ATAC-Acc by 29%29\%. 3-mer Corr drops by 9%9\% and JASPAR Corr by less than 1%1\%. The temperatures and the guidance window drive naturalness. A larger candidate count gives more direct help to utility. M=32M=32 provides a strong per-step default within the ablation sweep. The final held-out DRAKES Q row uses a larger budget, M=52M=52 with K=8K=8, for the no-budget-limit quality comparison in Table 2. L.5 Completion Count and Matched-state Proxy Diagnostics Completion count nhn_h and candidate count M address different aspects of sampling: M controls candidate-state coverage, whereas nhn_h controls the number of prediction-head completions averaged for each candidate in (3.5). We reserve N for sequence length. Table 16 reports the full sampler with five seeds and 3,2003,200 sequences per setting. Increasing nhn_h from 11 to 22 raises final guidance reward by 0.4920.492, at 41%41\% additional wall time; nh=3n_h=3 raises reward by 0.7060.706, at 81%81\% additional wall time. We retain nh=1n_h=1 as the practical compute–quality default. More completions reduce sampling variance for the head expectation, but do not remove its factorization or temperature mismatch with sequential rollouts. The completion-count sweep therefore measures a practical quality–cost trade-off while retaining the same factorized head approximation. Table 16: Completion-count ablation on HepG2. We use five seeds and 3,2003,200 sequences per setting. The score is final guidance reward R, not held-out Pred-Activity. Runtime increases are relative to nh=1n_h=1. Completions nhn_h Final guidance reward R Wall-time increase 11 5.272±0.0575.272± 0.057 baseline 22 5.764±0.0335.764± 0.033 41%41\% 33 5.978±0.0265.978± 0.026 81%81\% To examine the completion law, we save 640640 matched partial states at each of three stages from 128128 DRAKES HepG2 source trajectories per seed over five seeds. From each state, we independently generate 3232 direct prediction-head completions and 3232 sequential reverse-process continuations. For each route, we compute the h-value corresponding to each completion’s reward and average these values over the 3232 completions. Table 17 summarizes the pooled distributions of these per-state averages. The fixed-base fraction is the percentage of nucleotides already revealed in the partial state. W1 is Wasserstein-1 distance; KS is the maximum absolute difference between the two empirical cumulative distribution functions, not a significance-test result. Table 17: Matched-state completion diagnostic. At each stage, 640640 partial states come from 128128 source trajectories per seed over five seeds. Each state is independently completed 3232 times by the direct prediction head and 3232 times by sequential continuation. Moments and distances summarize the pooled distributions of per-state average h-values; parentheses denote variance, not uncertainty. Fixed bases Direct: mean (variance) Rollout: mean (variance) W1 KS 0.78%0.78\% 3.494(0.974)3.494\ (0.974) 3.480(1.097)3.480\ (1.097) 0.0600.060 0.0450.045 33.39%33.39\% 3.536(1.241)3.536\ (1.241) 3.611(1.109)3.611\ (1.109) 0.1010.101 0.0500.050 66.34%66.34\% 3.648(1.685)3.648\ (1.685) 3.772(1.620)3.772\ (1.620) 0.1240.124 0.0580.058 The pooled means are similar, while W1 increases from 0.0600.060 at 0.78%0.78\% fixed bases to 0.1240.124 at 66.34%66.34\%. KS remains between 0.0450.045 and 0.0580.058. Thus the direct head gives a useful low-cost marginal approximation in this diagnostic, with residual distributional shift that grows across the sampled stages. These pooled summaries do not imply equality of the conditional completion laws or exact sampling from the reward-tilted target. L.6 Soft Selection versus Argmax Table 18 compares soft selection in (3.7) with direct argmax using M=8M=8, τ=3τ=3, K=4K=4, and β=0.2β=0.2, over five seeds and 3,2003,200 sequences per setting. This diagnostic is separate from the high-budget Q operating point. As ω rises from 11 to 88, mean ESS falls from 1.8771.877 to 1.1431.143 and top-candidate selection rises from 77.53%77.53\% to 95.72%95.72\%. At ω=1ω=1, soft selection has lower final guidance reward than argmax (4.894±0.0554.894± 0.055 versus 5.154±0.0275.154± 0.027); at ω=8ω=8, it has higher reward (5.237±0.0515.237± 0.051). The behavior is therefore close to stochastic best-of-M selection, rather than broad resampling, but not identical to argmax. The remaining 4.28%4.28\% non-top selections at ω=8ω=8 can change subsequent states and candidate pools. Low ESS describes local weight concentration, not final-sequence diversity or target-distribution accuracy. Table 18: Soft selection versus direct argmax on HepG2. We use M=8M=8, τ=3τ=3, K=4K=4, and β=0.2β=0.2, with five seeds and 3,2003,200 sequences per setting. ESS and top-candidate frequency describe guided-step selection; rewards are final guidance reward R. Direct argmax does not depend on ω. ω Mean ESS Largest normalized weight Top-candidate selection rate Soft reward R Argmax reward R 11 1.877±0.0201.877± 0.020 0.7750.775 77.53%77.53\% 4.894±0.0554.894± 0.055 5.154±0.0275.154± 0.027 44 1.237±0.0041.237± 0.004 0.9270.927 92.69%92.69\% 5.183±0.0385.183± 0.038 5.154±0.0275.154± 0.027 88 1.143±0.0021.143± 0.002 0.9570.957 95.72%95.72\% 5.237±0.0515.237± 0.051 5.154±0.0275.154± 0.027 L.7 Split Time and Reward Trajectories Earlier branching improves reward at greater cost. Table 19 varies only ssplits_split, keeping M=9M=9, K=4K=4, and nh=1n_h=1 fixed. Splitting at update 00 gives final guidance reward 5.890±0.0285.890± 0.028 and a best-of-44 gain of 1.605±0.0191.605± 0.019; splitting at update 105105 gives 5.272±0.0575.272± 0.057 and 0.959±0.0200.959± 0.020, respectively. The earlier split costs 793.6±2.2793.6± 2.2 seconds per job, compared with 326.4±0.5326.4± 0.5 seconds for the late split. Late branching retains a useful selection gain while sharing more of the trajectory computation, but it cannot revise tokens already fixed before the split. Table 19: Controlled split-position ablation with M=9M=9, K=4K=4, and nh=1n_h=1 fixed. Only ssplits_split varies. Both final reward and best-of-44 selection gain use the guidance reward R, not held-out Pred-Activity. Split update ssplits_split Final guidance reward R Best-of-44 gain Wall time/job (s) 00 5.890±0.0285.890± 0.028 1.605±0.0191.605± 0.019 793.6±2.2793.6± 2.2 3232 5.855±0.0355.855± 0.035 1.545±0.0301.545± 0.030 653.8±1.5653.8± 1.5 6464 5.729±0.0295.729± 0.029 1.435±0.0191.435± 0.019 510.4±1.9510.4± 1.9 105105 5.272±0.0575.272± 0.057 0.959±0.0200.959± 0.020 326.4±0.5326.4± 0.5 Diversity remains in the unresolved suffix. For saved partial states from the frozen reverse process, we sample 3232 independent suffix completions and compute pairwise Hamming distance only on positions that were still masked (Table 20). The distance is 0.710±0.0100.710± 0.010 with 25.43%25.43\% of bases fixed, and remains 0.661±0.0110.661± 0.011 with 90.38%90.38\% fixed. These results demonstrate substantial suffix diversity conditional on a shared prefix, not determinism of early reveals. Late best-of-K exploits this remaining search space; the complete D-DOIT–TDS comparison does not isolate branching’s contribution. Table 20: Suffix diversity under the frozen reverse process. For each saved partial sequence, 3232 independent suffix completions are sampled. Hamming distance is the mean pairwise fraction of differing bases restricted to positions that were still masked in the shared partial state. Fixed positions Still-masked positions Remaining-position Hamming distance 25.43%25.43\% 74.57%74.57\% 0.710±0.0100.710± 0.010 50.38%50.38\% 49.62%49.62\% 0.689±0.0100.689± 0.010 90.38%90.38\% 9.62%9.62\% 0.661±0.0110.661± 0.011 Completion-proxy progress is nonmonotone. Table 21 scores partial states after filling masked positions with the frozen model’s deterministic argmax predictions. The held-out Enformer scores are averaged over 640640 sequences per seed and then over five seeds. They are neither guidance reward R nor the batch-median Pred-Activity metric. D-DOIT’s completion-proxy score rises from 2.0672.067 at update 11 to 2.3282.328 at update 3232, falls to 1.7581.758 at update 9696, and reaches 3.6703.670 at update 128128; frozen sampling declines from 2.0532.053 to 0.3710.371. This trajectory shows higher endpoint activity without monotone intermediate progress. An argmax-completion score is not the rollout h-function or a guarantee about future reward. Table 21: Held-out enformer completion-proxy trajectory. At each update, masked positions are filled with the frozen model’s deterministic argmax predictions before scoring. Scores are averaged over 640640 sequences per seed and then over five seeds, rather than using the batch-median Pred-Activity aggregation. Update 11 3232 6464 9696 128128 D-DOIT 2.0672.067 2.3282.328 1.7631.763 1.7581.758 3.6703.670 Frozen sampling 2.0532.053 1.5611.561 1.0581.058 0.7160.716 0.3710.371 From update 3232 onward, the argmax-completed D-DOIT sequences are 100%100\% unique, and their mean pairwise Hamming distance increases from 0.7260.726 to 0.7490.749. This diversity measure concerns completed sequences across the sample batch, unlike the remaining-position conditional diversity above. The observed late-stage reward improvement is therefore compatible with maintained diversity in this diagnostic, rather than implying that guidance universally preserves diversity. Appendix M Experimental Details of Protein Inverse Folding Protein inverse folding tests amino-acid sequence design conditional on a target backbone, complementing the regulatory DNA experiments, and the full cross-method comparison, including all metrics and seed-level SDs, is in Table 22. We follow the protein inverse-folding protocol of DRAKES [8], using the same reward oracle and evaluation metrics. D-DOIT guides the frozen conditional generator through reward-value queries without updating its parameters. Protocol and aggregation. We evaluate all 1212 test protein backbones with three independent sampling seeds and generate 128128 sequences per backbone and seed. The configuration M=52M=52, K=8K=8, and nh=1n_h=1 (one completion per candidate) is fixed before test-set evaluation, with no tuning after inspecting test results. Each metric is computed separately for each seed under the DRAKES protocol; D-DOIT entries report the mean and sample standard deviation across those three seed-level metrics. The uncertainty describes sampling-seed variation, not variation across sequences or backbones. Metrics. Pred-ddG is the predicted stability score, with higher values preferred under the benchmark’s sign convention. We report its median and the percentage of sequences with predicted ddG greater than zero. Structural fidelity is measured by self-consistency RMSD (scRMSD), comparing the predicted structure of a generated sequence with its target backbone; we report the median in Å and the percentage below 22 Å. Joint success requires both positive predicted ddG and scRMSD below 22 Å. These are predictor-based measures rather than experimental stability or folding measurements. Table 22: Protein inverse folding on the DRAKES benchmark. D-DOIT entries are mean ± sample SD across seed-level metrics. Other rows and uncertainties are from DRAKES Table 2 [8], not rerun. Bold marks the best mean per column, not statistical significance. Method Pred-ddG median ↑ ddG >0>0 (%) ↑ scRMSD median (Å) ↓ scRMSD <2<2 Å (%) ↑ Success (%) ↑ Pre-trained −0.544±0.037-0.544± 0.037 36.6±1.036.6± 1.0 0.849±0.0130.849± 0.013 90.9±0.690.9± 0.6 34.4±0.534.4± 0.5 CG −0.561±0.045-0.561± 0.045 36.9±1.136.9± 1.1 0.839±0.0120.839± 0.012 90.9±0.690.9± 0.6 34.7±0.934.7± 0.9 SMC 0.659±0.0440.659± 0.044 68.5±3.168.5± 3.1 0.841±0.0060.841± 0.006 93.8±0.493.8± 0.4 63.6±4.063.6± 4.0 TDS 0.674±0.0860.674± 0.086 68.2±2.468.2± 2.4 0.834±0.0010.834± 0.001 94.4±1.294.4± 1.2 62.9±2.862.9± 2.8 CFG −1.186±0.035-1.186± 0.035 11.0±0.411.0± 0.4 3.146±0.0623.146± 0.062 29.4±1.029.4± 1.0 1.3±0.41.3± 0.4 D-DOIT 0.870±0.0460.870± 0.046 74.8±3.174.8± 3.1 0.872±0.0090.872± 0.009 92.5±1.392.5± 1.3 68.1±3.568.1± 3.5 Results and comparison boundary. D-DOIT achieves the highest median Pred-ddG (0.870±0.0460.870± 0.046), ddG-positive fraction (74.8±3.1%74.8± 3.1\%), and joint success (68.1±3.5%68.1± 3.5\%) among the six methods. Joint success exceeds the published SMC and TDS means by 4.54.5 and 5.25.2 percentage points, respectively. TDS retains better structural fidelity: median scRMSD is 0.8340.834 Å versus 0.8720.872 Å, and the fraction below 22 Å is 94.4%94.4\% versus 92.5%92.5\%. All non-D-DOIT entries are taken directly from DRAKES Table 2 and were not rerun in this comparison; their reported uncertainties are retained. The shared benchmark oracle and metrics permit a literature comparison, not a matched-compute or matched-implementation claim. The results extend the computational evaluation to protein sequence design; they do not establish experimentally measured stability or folding. Appendix N Licenses for Existing Assets We use pre-existing assets for non-commercial academic research and credit their creators through Sections J and K. The Masked Discrete Diffusion Model reference implementation of Sahoo et al. [3] carries Apache-2.0; the Ctrl-DNA repository of Chen et al. [25], Enformer implementation of Avsec et al. [64], and HyenaDNA repository of Nguyen et al. [55] also carry Apache-2.0 licenses. The HepG2 enhancer corpus from Gosai et al. [24] is distributed through the paper’s data-availability resources, and the Nature article carries C BY 4.0. The ENCODE Data Use Policy governs chromatin-accessibility data [65]; JASPAR 2022 [66] is open access and its Bioconductor data package is GPL-2. The public DRAKES repository associated with Wang et al. [8] had no top-level software license in the GitHub metadata at access time. We use DRAKES only as a cited benchmark, baseline, oracle, and checkpoint source for academic evaluation, and do not redistribute its code or checkpoints.