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Risk-Calibrated Proposal Transport for Finite-Particle Diffusion Steering
Ziseok Lee, Jaehyeon Kim, Seungwon Kim, Seunghyun Moon, Haneul Choi, Wooyeol Lee, Donghyun Koh, Minhyeong Lee, Kyungsu Kim
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Summary
The paper introduces Risk-Calibrated Proposal Transport (RCPT), a method to mitigate the deployment risks of Variance-Controlling Guidance (VCG) in finite-particle diffusion steering. While VCG minimizes fitting residuals, it can increase out-of-fit risk and cause particle collapse. RCPT uses deletion leave-one-out residuals to calibrate the retained fraction of the VCG update, balancing population headroom against estimation penalty. Experiments on synthetic distributions, molecular scaffold decoration, and CIFAR-10 generation show RCPT recovers from harmful updates and improves terminal metrics compared to uncalibrated VCG and other baselines.
Entities (9)
Relation Signals (7)
RCPT → evaluatedon → CIFAR-10
confidence 95% · Experiments on ... class-conditional CIFAR-10 generation demonstrate recovery from harmful fitted updates.
RCPT → mitigates → VCG
confidence 95% · RCPT mitigates harmful fitted updates and improves a broad range of terminal metrics relative to uncalibrated VCG.
RCPT → uses → deletion leave-one-out residuals
confidence 95% · RCPT uses deletion leave-one-out residuals to calibrate the retained fraction of the VCG update
VCG → cancause → particle collapse
confidence 90% · The resulting update can degrade unweighted generation or accelerate particle collapse.
FKC → compensatesfor → proposal mismatch
confidence 90% · Feynman-Kac correction compensates for proposal mismatch through importance-weighted sequential Monte Carlo (SMC)
RCPT → evaluatedon → QM9
confidence 90% · Multi-Property Molecular Reward Steering on QM9
RCPT → evaluatedon → CrossDocked2020
confidence 90% · Scaffold Decoration on CrossDocked2020
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Abstract
Abstract:Inference-time steering combines pretrained diffusion experts or rewards without retraining by changing the dynamics that transport noise to data. Feynman-Kac correction compensates for proposal mismatch through importance-weighted sequential Monte Carlo (SMC), whose finite-particle behavior depends on the proposal. Variance-controlling guidance (VCG) improves that proposal by fitting a linear drift correction to minimize empirical log-weight-rate variance. Although its population optimum cannot worsen residual variance, finite-particle VCG can nearly eliminate its fitting residual while increasing residual risk on new states by orders of magnitude. The resulting update can degrade unweighted generation or accelerate particle collapse. We show that the centered Feynman-Kac rate is the normalized transport residual and that expected out-of-fit benefit is exactly population headroom minus coefficient-estimation penalty. Under regularity assumptions, a Wasserstein analysis bounds the unweighted proposal's terminal error using this residual. These results motivate Risk-Calibrated Proposal Transport (RCPT), which uses deletion leave-one-out residuals to calibrate the retained fraction of the VCG update, adding no model calls and only small linear-algebra overhead. Experiments on 2D checker distributions, scaffold decoration, molecular property optimization, and class-conditional CIFAR-10 generation demonstrate recovery from harmful fitted updates. Across molecular and image domains, RCPT mitigates harmful fitted updates and improves a broad range of terminal metrics relative to uncalibrated VCG.
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- Source: https://arxiv.org/abs/2610.04171v1
- Canonical: https://arxiv.org/abs/2610.04171v1
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Risk-Calibrated Proposal Transport for Finite-Particle Diffusion Steering Ziseok Lee 1,† , Jaehyeon Kim 1 , Seungwon Kim 2,1 , Seunghyun Moon 2,1,† , Haneul Choi 2,1 , Wooyeol Lee 1 , Donghyun Koh 2,1 , Minhyeong Lee 2,1,† , Kyungsu Kim 1,† 1 Seoul National University 2 Asteromorph † corresponding authors: ziseoklee, minhyeong.lee, kyskim@snu.ac.kr, sh.moon@asteromorph.com Inference-time steering combines pretrained diffusion experts or rewards without retraining by changing the dynamics that transport noise to data. Feynman–Kac correction compensates for proposal mismatch through importance-weighted sequential Monte Carlo (SMC), whose finite-particle behavior depends on the proposal. Variance-controlling guidance (VCG) improves that proposal by fitting a linear drift correction to minimize empirical log-weight-rate variance. Although its population optimum cannot worsen residual variance, finite-particle VCG can nearly eliminate its fitting residual while increasing residual risk on new states by orders of magnitude. The resulting update can degrade unweighted generation or accelerate particle collapse. We show that the centered Feynman–Kac rate is the normalized transport residual and that expected out-of-fit benefit is exactly population headroom minus coefficient-estimation penalty. Under regularity assumptions, a Wasserstein analysis bounds the unweighted proposal’s terminal error using this residual. These results motivate Risk-Calibrated Proposal Transport (RCPT), which uses deletion leave-one-out residuals to calibrate the retained fraction of the VCG update, adding no model calls and only small linear-algebra overhead. Experiments on 2D checker distributions, scaffold decoration, molecular property optimization, and class-conditional CIFAR-10 generation demonstrate recovery from harmful fitted updates. Across molecular and image domains, RCPT mitigates harmful fitted updates and improves a broad range of terminal metrics relative to uncalibrated VCG. Lab: https://aibl.snu.ac.kr 1 Introduction Inference-time steering adapts pretrained diffusion or flow models to targets assembled from experts or rewards. These models transport noise toward data through a stochastic differential equation (SDE) [39], dX t = v t (X t ) dt + σ t dW t , X 0 ∼ p 0 , wherev t is the drift,σ t is the noise amplitude, andW t is Brownian motion. An ordinary differential equation (ODE) sampler uses zero noise and an appropriate drift; steering changes the drift while keeping the pretrained models fixed. A cheap guided proposal need not follow the requested target path. Combining expert fields, as in classifier-free guidance [19], gives an unweighted Anchor, but matching an instantaneous score does not ensure that the prescribed marginals satisfy the sampler’s forward equation. Feynman–Kac correction (FKC) and Adaptive Path Correction with Exponents (ACE) account for this mismatch through importance weights and SMC [26,36]. Their target recovery is asymptotic: finite-particle resampling can deplete ancestral diversity, while unweighted deployment retains proposal bias. Proposal fitting aims to improve both deployment modes by estimating a drift correction from an ensemble of samples, or particles. DriftLite’s variance-controlling guidance (VCG) minimizes FK log-weight-rate variance in a linear field class, while energy-controlling guidance (ECG) fits gradient fields through an energy objective [33]. Population VCG 1 arXiv:2610.04171v1 [stat.ML] 3 Oct 2026 cannot increase residual variance relative to Anchor because its class includes the zero correction. With finite particles, that benefit depends on whether the estimated coefficients generalize. Limited fitting information can reverse this benefit. We focus on budgets below 100 fitting particles, with larger populations testing how calibration responds as information grows. When effective information approaches controller rank, VCG can nearly eliminate its fitting residual yet increase independent residual risk by orders of magnitude (Figure 1). Increasing the fitting population costs additional expert evaluations, motivating calibration of the available update. We treat proposal fitting as a prediction problem whose benefit must survive deployment. The centered FK rate is the normalized transport residual, and expected out-of-fit improvement equals population headroom minus coefficient- estimation penalty. This separates the improvement available to an ideal controller from its estimation cost. A Wasserstein analysis bounds unweighted terminal proposal error through the residual, which also controls leading local correction-weight inflation. Risk-Calibrated Proposal Transport (RCPT) estimates this tradeoff and retains a scalar fraction of the VCG update. Deletion leave-one-out residuals estimate the residual scale, while a structured penalty accounts for controller complexity relative to fitting information. RCPT adds no model calls and only small linear-algebra overhead to VCG proposal fitting. It supports frozen or online deployment and retains more of VCG as fitting information grows. Across molecular and image domains, RCPT mitigates harmful fitted updates and improves terminal metrics relative to baselines including uncalibrated VCG (Tables 3–5). Our contributions are: •A finite-particle failure mode hidden by population guarantees. We formulate proposal fitting as an out-of-fit prediction problem and expose fit-to-deployment inversion: empirical VCG can nearly eliminate fitting residual while increasing independent risk and degrading unweighted or corrected generation. We connect the centered FK residual to a Wasserstein terminal-error bound and leading local importance-weight inflation. •Deployment-risk theory and nearly cost-free calibration. We derive an exact headroom–penalty decomposition, an asymptotic estimation penalty, and a bias-aware scalar oracle. RCPT instantiates this principle with structured deletion-LOO calibration, no additional expert calls, and a conditional bound on its decision error. •Deployment-scale evidence across inference modes and domains. Controlled 10,000-particle deployments and common-fit regularization audits test generalization directly. A complete 76-pocket scaffold benchmark, eight-seed molecular SMC protocols, and CIFAR-10 generation show that calibration repairs harmful VCG fits and improves downstream quality, with restored genealogy under corrected molecular sampling. 10 −10 10 −5 10 0 Fitting risk / Anchor 10 0 10 2 10 4 Independent risk / Anchor ↓ VCG 10 −10 10 −5 10 0 Fitting risk / Anchor RCPT 4 16 64 256 N fit Lower independent risk = better deployment Figure 1. Low fitting risk can hide high deployment risk. Lower on the vertical axis means better independent risk; a smaller fitting residual alone is insufficient. Both axes show log-scaled ratios to Anchor; dotted lines mark Anchor. Color denotes fitting population. 2 2 Related Work Diffusion steering and Anchor proposals.Score composition and reward guidance construct cheap proposals from pretrained models [2,6,9,13,19,38,49,50]. FKC and ACE correct prescribed paths through importance weighting and SMC [12,26,28,36]. Related particle methods support diffusion sampling, composition, and discrete correction [17,24,35,42,46,47]. Finite-particle costs and ancestral collapse motivate improving the proposal before correction [1,5,8,25]. RCPT keeps the target fixed and calibrates a fitted update to Anchor; Table 2 compares both unweighted and SMC deployment. Proposal fitting: VCG and ECG. Our closest baselines are DriftLite’s VCG, which minimizes FK rate variance, and ECG, which fits gradient corrections through an energy objective [33]. Proposal fitting improves sampling through adaptive importance sampling, controlled or twisted SMC, and diffusion or incremental-divergence objectives [10,18,23,31,37,43,45]. Optimal discrete proposals, tree sampling, trust-region twists, and guidance-variance control offer complementary mechanisms [21,22,29,44]; biased-proposal theory separates approximation and particle errors [41]. RCPT addresses the estimation error of VCG’s fitted coefficient; Table 2 compares VCG, ECG, and RCPT on the same Checker deployments. Regression regularization and calibration.Regularized and selected Stein control variates address unstable regres- sions [27,40], while classical shrinkage and covariance penalties quantify prediction error [14,15]. RCPT’s contribution is to identify and control the deployment risk of a regression coefficient that changes generative dynamics. Ridge penalizes coefficients and principal-component regression (PCR) truncates fitted directions; RCPT instead calibrates one amplitude using structured headroom and penalty estimates. Direct LOO substitutes a cross-validated amplitude for this structured rule and serves as a calibration ablation. Table 1 and the paired comparisons in Section 6 show lower mean reference risk for RCPT than Direct LOO and LOO-selected ridge/PCR in the common-fit Checker audit. 3 Finite-Particle Proposal Risk Target path and Markov proposal. Letep t > 0be a differentiable unnormalized density,Z t = R ep t (x) dx, and p t = ep t /Z t fort∈ [0, 1]. In the applications, whereq (i) t are expert densities,R j,t are reward functions, andγ i (t),β j,t are time-varying exponents, the target path is ep t (x) = I Y i=1 q (i) t (x) γ i (t) exp J X j=1 β j,t R j,t (x) ,(1) which includes products, ratios, time-varying exponents, and reward tilting. LetL ∗ t,v be the conservative forward Kolmogorov operator of a proposal with selectable driftv t , and letU v s,t be its Markov evolution on observables. The unweighted proposal law ρ v t satisfies ∂ t ρ v t =L ∗ t,v ρ v t and ρ v 0 = p 0 . The corresponding raw Feynman–Kac rate is g v t (x) = ∂ t ep t (x)−L ∗ t,v ep t (x) ep t (x) .(2) Proposal-Only deployment discards this correction; FKC/ACE retains its incremental weights and a stated resampling rule. RCPT changes neither the target path nor the exponent schedule. Population VCG is a projection. For Anchor driftv 0 t andmfixed fieldsφ j,t , fit an additive driftv λ t = v 0 t + P m j=1 λ j,t φ j,t . With unchanged diffusion, the FK rate becomesg λ t = g 0 t + a ⊤ t λ t , wherea j,t = T p t φ j,t andT p t φ = p −1 t ∇·(p t φ)is the Stein transport operator. Define the population residual riskV t (λ) = Var p t (g λ t ),C t = Cov p t (a t ,a t ), andd t = Cov p t (a t ,g 0 t ). The covariance-weighted seminorm∥z∥ 2 C t = z ⊤ C t z = Var p t (a ⊤ t z)measures how much a coefficient change affects the residual. This is theC t geometry: errors are weighted by their effect on residual variance, rather than by their Euclidean size. The minimum-norm population VCG coefficient isλ ⋆ t =−C † t d t , where†denotes the Moore–Penrose pseudoinverse [33]. Writing V t,⋆ = V t (λ ⋆ t ) gives V t (λ) = V t,⋆ +∥λ− λ ⋆ t ∥ 2 C t .(3) Appendix D.1 gives the derivation, including singular C t . 3 3.1 Headroom and coefficient-estimation error We define out-of-fit risk by evaluating on the target law and averaging over fitting samples. At timet, sampleD produces b λ t (D)with expected prediction riskR t ( b λ t ) = E D V t ( b λ t (D)). ThusV t evaluates a fixed coefficient, whileR t also averages its estimation variability. We now fix t and omit time subscripts in the coefficient-risk analysis below. Theorem 1 (Exact out-of-fit risk decomposition). For a fixed target law and field dictionary, assumeE D ∥ b λ∥ 2 C <∞. Define population headroom H =∥λ ⋆ ∥ 2 C and coefficient-estimation penalty P = E D ∥ b λ− λ ⋆ ∥ 2 C . Then R( b λ)− V (0) = P − H.(4) Taking expectation in Eq. (3) proves the identity. The penalty includes both coefficient variance and squared bias, so it does not require unbiased fitting. The full estimated update harms expected residual risk precisely whenP > H. Here, estimation error exceeds the available improvement. 3.2 Scaling the fitted correction to minimize risk Scaling b λ byα ∈ [0, 1]interpolates between Anchor and the full VCG fit. The optimal fixed amplitude balances retained signal against estimation error. Theorem 2 (Expected-risk oracle and the structured case). LetB = (λ ⋆ ) ⊤ CE D ( b λ− λ ⋆ )measure alignment between mean estimation error and the population coefficient. For deterministic α∈ [0, 1], R(α b λ) = V ⋆ + (1− α) 2 H + α 2 P − 2α(1− α)B.(5) If H + P + 2B > 0, the minimizing amplitude is α ⋆ = Π [0,1] H + B H + P + 2B ,(6) whereΠ [0,1] clips to[0, 1]; ifH + P + 2B = 0, all amplitudes have the same risk. WhenB = 0andH + P > 0, this becomes α 0 = H/(H + P ), with R(α 0 b λ)− V (0) =− H 2 H + P .(7) The oracle retains more of the fit when headroom is large relative to estimation error andB = 0. Unbiased fitting is sufficient forB = 0; more generally, only its mean error’s alignment withλ ⋆ underCmust vanish. The fraction H/(H + P )motivates RCPT’s structured rule. Section 4 analyzes data-selected amplitudes, and Appendix D.5 quantifies sensitivity to nonzero B. 4 Risk-Calibrated Proposal Transport RCPT uses a structured estimate of prediction risk to select the strength of the fitted VCG correction. Candidate fit. Givena n ,g 0 n ,ω n N n=1 with0 ≤ ω n < 1and P n ω n = 1 , let ̄a = P n ω n a n , ̄g 0 = P n ω n g 0 n , and form the weighted centered moments b C and b d. The coefficient b λ =−( b C + ηI) † b d(8) minimizes P n ω n (g 0 n + a ⊤ n λ− c) 2 + η∥λ∥ 2 2 over (λ,c). This is the VCG fit reused by RCPT. Structured penalty and headroom. Delete observationn, renormalizeω (−n) i = ω i /(1− ω n ), and refit with the same ridge convention. With the deleted weighted means, e n = g 0 n + a ⊤ n b λ (−n) −bc (−n) , bc (−n) = ̄g 0 −n + ̄a ⊤ −n b λ (−n) .(9) 4 Algorithm 1 Risk-Calibrated Proposal Transport (structured H,P ) Input: target path, Anchor, basis, weighted ensemble, ridge η. 1. Fit b λ by Eq. (8). 2. Compute deletion residuals by Eq. (9). 3. Form b P , b H , and bα by Eq. (10). 4. Deploy v 0 + P j bα b λ j φ j online, or freeze the schedule for independent deployment. Set bσ 2 LOO = P n ω n e 2 n , b N eff = ( P n ω 2 n ) −1 , br eff = tr[ b C( b C + ηI) † ], and S = b λ ⊤ b C b λ. RCPT uses b P = bσ 2 LOO br eff b N eff , b H = [S− b P ] + , bα = b H b H + b P + ε num , e λ = bα b λ,(10) whereε num > 0stabilizes a zero denominator. The rank-over-information penalty is motivated by the classical regression expansionP = N −1 tr(ΩC † ) + o(N −1 ), which reduces toσ 2 ε rank(C)/NwhenΩ = σ 2 ε C(Lemma D.1, with proof in Appendix D.6). The effective-rank and normalized-weight factors give a scalar approximation to this covariance penalty. They retain a low-dimensional decision when the fitting ensemble is small. The criterion implemented by RCPT. Equation (10) is exactly bα = arg min 0≤α≤1 b ∆ S (α), b ∆ S (α) = (S + ε num )α 2 − 2(S− b P )α.(11) This quadratic gives a direct operational meaning to the rule.Smeasures the fitted correction’s empirical variation, and b Pdiscounts the supported improvement using deleted prediction residuals. The positive part returns Anchor when the estimated penalty exhausts the empirical gain. As supported gain grows relative to penalty, the rule retains more of the update. Deployment and computation. A frozen schedule can be reused on independent trajectories, while an online controller recalibrates at each fitted step. Proposal-Only sampling uses the selected drift directly; corrected SMC additionally retains FK weights and resampling. The implementation uses the same feature values and divergences as VCG. Additional work consists of small dense deletion-refit systems. Coefficient clipping and numerical fallbacks are specified separately in Appendix H. The fitted correction defines a quadratic risk along its direction. RCPT approximates this quadratic usingSand b P, and their errors control the selected risk. Theorem 3 (Calibration error of the implemented rule). Freeze a realized fit with finite residual second moments and write f = a ⊤ b λ, q = Var p (f ), and κ = Cov p (g 0 + f,f ). If|S− q|≤ e q and| b P − κ|≤ e κ , then RCPT satisfies V (bα b λ)− V (0)≤ e q + 2e κ ,(12) V (bα b λ)− min α∈[0,1] V (α b λ)≤ 2e q + 4e κ + ε num .(13) Forq > 0, the ray oracle isα ray = Π [0,1] (1−κ/q); when it is interior, excess risk isq(bα−α ray ) 2 (Appendix D.4). This curvature-weighted regret measures amplitude-selection error. The Checker audit evaluates it against an independent corrected-SMC reference law (Appendix I.2). 4.1 Residual risk upper-bounds Wasserstein error The centered FK rate is the source that separates the proposal from the target path. Its minimum correction energy preserves the spatial geometry of this source and gives a direct W 2 bound. Theorem 4 (FK residual and Wasserstein transport). Under Appendix C.1’s conservative evolution and regularity assumptions, definer v t = g v t − E p t g v t . Then∂ t logZ t = E p t g v t and∂ t p t =L ∗ t,v p t + p t r v t . Suppose additionally that 5 the diffusion onR d is state-independent, the laws have finite second moments, and⟨x−y,v t (x)−v t (y)⟩≤ L t ∥x−y∥ 2 withL∈ L 1 (0, 1). Let−∇· (p t ∇ψ t ) = p t r v t have finite energyE t (v) = E p t ∥∇ψ t ∥ 2 , with its corrected dynamics well posed. For ρ v 0 = p 0 , W 2 (p 1 ,ρ v 1 )≤ Z 1 0 K v (t, 1) p E t (v) dt, K v (t, 1) = exp Z 1 t L s ds .(14) If p t has Poincaré constant C P (t), thenE t (v)≤ C P (t)V t (v), where V t (v) = E p t (r v t ) 2 . The proof in Appendix C.3 adds the minimum-energy velocity that removes the residual and couples the corrected and proposal processes with common noise. The integral retains the residual’s time profile before the optional Poincaré conversion to variance. This specializes the transport geometry of Benamou and Brenier[4], Peyre[30]to fitted FK proposals. The same residual also controls the leading local cost of FK weighting. Proposition 1 (Local correction-weight second moment). Initialize a proposal segment atX t ∼ p t . LetW v t,∆ = dP v t,∆ / dQ v t,∆ be its exact Feynman–Kac segment weight, normalized to mean one. IfW v t,∆ = 1 + ∆r v t (X t ) +o L 2 (∆), then E Q [( W v t,∆ ) 2 ] = 1 + χ 2 (P v t,∆ ∥Q v t,∆ ) = 1 + ∆ 2 V t (v) + o(∆ 2 ).(15) The expansion follows by squaring the normalized local weight and taking expectation (Appendix C.4). Residual risk therefore has two concrete roles: it controls a transport upper bound for unweighted deployment and the leading local correction-weight variance for corrected sampling. We evaluate terminal discrepancies and genealogy separately to measure this behavior. 5 Experiments Evaluation Design. We separate fitting from evaluation and never use terminal metrics to select a controller. Frozen schedules deploy on fresh trajectories; online controllers refit the live population. Common-state risk compares methods on shared evaluation states, whereas active-path risk follows each method’s trajectory. The four experiments test independent risk, terminal generation, and particle genealogy; Appendix I gives implementation details and cohort definitions. 5.1 Synthetic Checker Distribution Table 1. Independent risk on common Checker fits. Mean integrated Risk/Anchor over ten paired fits; lower is better. Direct LOO ablates calibration; ridge and PCR select regularization. N fit AnchorVCGDirect LOOLOO ridgeLOO PCRRCPT 81.0000244.50.92390.92111.03000.8013 161.0000174.01.12600.83061.17400.4931 321.00002.7970.38210.37050.61270.3589 Table 2. Checker transport. Ten paired seeds, fit 16/deploy 2,000; all endpoints are finite. VCG/RCPT use frozen schedules and ECG the raw configuration. Lineages counts terminal ancestors among 2,000 SMC particles. Bold marks the best mean; Appendix Table I.2 ablates calibration. Proposal-OnlyCorrected SMC Controller W 1 ↓ W 2 ↓ MMD↓ W 1 ↓ W 2 ↓ MMD↓ Lineages↑ Anchor0.474 0.6460.0480.367 0.5100.048338.7 VCG0.814 1.0190.2262.637 3.0335.123381.6 ECG1.394 1.6360.3250.833 1.1130.214233.7 RCPT0.4420.5750.0730.3160.4640.042568.5 6 Checker isolates finite-particle fitting from independent deployment. A common-fit audit uses 12 fields,N = 8, 16, 32, and ten paired fits per budget; independent corrected-SMC reference systems evaluate residual risk, and 10,000 fresh trajectories per fit test deployment. A separate matched cohort fits 16 particles and deploys 2,000 per method in both Proposal-Only and SMC modes. Appendix I.2 details the reference construction and common-fit replay, and Appendix I specifies the matched endpoint cohort. RCPT has the lowest mean reference risk at every fitting budget (Table 1) and lower meanW 1 andW 2 than Anchor, VCG, and ECG in both matched deployment modes (Table 2). Under SMC, it also has the lowest MMD and retains 568.5terminal lineages, versus338.7for Anchor. Proposal-Only MMD favors Anchor, illustrating that endpoint rankings depend on the metric. Figure 2 visualizes generated distributions; Appendix Table I.4 reports common-fit endpoints and completion counts. Proposal-Only Target Target Anchor W 2 =0.769, MMD=0.065 VCG W 2 =1.193, MMD=0.257 RCPT (Ours) W 2 =0.682, MMD=0.067 ECG W 2 =1.316, MMD=0.472 SVGD W 2 =0.604, MMD=0.043 Anchor Target Anchor W 2 =0.769, MMD=0.065 VCG W 2 =1.193, MMD=0.257 RCPT (Ours) W 2 =0.682, MMD=0.067 ECG W 2 =1.316, MMD=0.472 SVGD W 2 =0.604, MMD=0.043 W2 0.769 MMD 0.065 VCG Target Anchor W 2 =0.769, MMD=0.065 VCG W 2 =1.193, MMD=0.257 RCPT (Ours) W 2 =0.682, MMD=0.067 ECG W 2 =1.316, MMD=0.472 SVGD W 2 =0.604, MMD=0.043 W2 1.193 MMD 0.257 ECG Target Anchor W 2 =0.769, MMD=0.065 VCG W 2 =1.193, MMD=0.257 RCPT (Ours) W 2 =0.682, MMD=0.067 ECG W 2 =1.316, MMD=0.472 SVGD W 2 =0.604, MMD=0.043 W2 1.316 MMD 0.472 RCPT (Ours) Target Anchor W 2 =0.769, MMD=0.065 VCG W 2 =1.193, MMD=0.257 RCPT (Ours) W 2 =0.682, MMD=0.067 ECG W 2 =1.316, MMD=0.472 SVGD W 2 =0.604, MMD=0.043 W2 0.682 MMD 0.067 SMC Target TargetAnchorVCG RCPT (Ours) ECGSVGD Anchor TargetAnchorVCG RCPT (Ours) ECGSVGD W2 1.389 MMD 0.139 VCG TargetAnchorVCG RCPT (Ours) ECGSVGD W2 5.721 MMD 6.946 ECG TargetAnchorVCG RCPT (Ours) ECGSVGD W2 1.762 MMD 0.486 RCPT (Ours) TargetAnchorVCG RCPT (Ours) ECGSVGD W2 0.945 MMD 0.123 Figure 2. Separate Checker Proposal-Only replay with ECG. Frozen fit-16 schedules, 10,000 fresh trajectories, and no resampling. Labels use a fixed 2,000-sample endpoint audit. This illustrative replay is separate from Table 2. RCPT gives the lowestW 2 ; the corresponding SMC results are in Appendix Table I.1. 5.2 Scaffold Decoration on CrossDocked2020 We evaluate all 76 CrossDocked2020 scaffold–pocket tasks of Lee et al.[26]. Four expert paths supply fragment, ligand, conformation, and pocket information. Sixteen particles fit each 12-field schedule, which is frozen for 16 fresh paired Proposal-Only outputs. Chemistry and docking are evaluated afterward; unique useful yield counts distinct valid, scaffold-retaining molecules that beat the reference docking score per generation slot. Appendix I.3 specifies the models, fields, and evaluation protocol. RCPT restores useful molecular generation after the expanded VCG fit (Table 3). Validity rises from0.229to0.434 and unique useful yield from0.118to0.251, with paired 95% pocket-bootstrap improvement intervals[0.151, 0.260] and[0.086, 0.181]. A separate 64-output deployment of the same schedules reproduces the VCG repair across 4,864 candidates per arm (Appendix Table I.5). Figure 3 shows matched molecular examples. Table 3. Scaffold decoration. Means over 76 pockets, fit 16/deploy 16. Yields use fixed generation denominators; risk is active-path variance normalized by Anchor. Unique useful counts distinct valid, scaffold-retaining, reference-beating outputs. † ECG is a separate 76-pocket run. Controller Validity↑ Scaffold↑ Risk/Anchor↓ Dock success↑ Affinity↑ Unique useful↑ Anchor0.4250.4161.0000.8900.2820.236 VCG0.2290.22526.2750.5690.1570.118 ECG † 0.2210.2153.5930.4650.1400.126 RCPT0.4340.4190.8770.9030.2970.251 5.3 Multi-Property Molecular Reward Steering on QM9 We steer pretrained EEGSDE [3] toward polarizability and HOMO–LUMO-gap targets on QM9 [32] using 12 gated reward-gradient fields. Each of eight split seeds fits on 16 particles and deploys eight fresh SMC batches of 16; eight online seeds instead refit 144 live particles. Independent property evaluators and canonical molecular identities measure attainment and diversity. Appendix I.4 gives the reward, controller, and resampling details. RCPT improves property attainment and genealogy in split deployment (Table 4): unique successful molecules increase from13.88for VCG to23.00per 128 outputs, while the surviving-lineage fraction rises from0.1611to0.3525. Online, 7 Anchor Scaffold AnchorVCGRCPT (Ours) QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.00 | QED 0.564 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -7.90 | QED 0.341 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.20 | QED 0.731 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.302 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.323 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.196 QVina -8.00 QED 0.564 QM9 AnchorVCGRCPT (Ours) QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.00 | QED 0.564 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -7.90 | QED 0.341 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.20 | QED 0.731 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.302 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.323 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.196 Error 0.302 VCG Scaffold AnchorVCGRCPT (Ours) QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.00 | QED 0.564 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -7.90 | QED 0.341 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.20 | QED 0.731 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.302 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.323 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.196 QVina -7.90 QED 0.341 QM9 AnchorVCGRCPT (Ours) QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.00 | QED 0.564 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -7.90 | QED 0.341 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.20 | QED 0.731 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.302 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.323 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.196 Error 0.323 RCPT (Ours) Scaffold AnchorVCGRCPT (Ours) QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.00 | QED 0.564 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -7.90 | QED 0.341 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.20 | QED 0.731 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.302 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.323 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.196 QVina -8.20 QED 0.731 QM9 AnchorVCGRCPT (Ours) QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.00 | QED 0.564 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -7.90 | QED 0.341 QVina: -8.00 QED:0.564 QVina: -7.70 QED:0.480 QVina: -7.90 QED:0.341 QVina: -8.00 QED:0.777 QVina: -8.20 QED:0.731 QVina: -8.30 QED:0.837 AnchorVCGRCPT (Ours) QVina -8.20 | QED 0.731 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.302 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.323 Error:0.302Error:0.344Error:0.323Error:0.145Error:0.196Error:0.066 AnchorVCGRCPT (Ours) Error 0.196 Error 0.196 Figure 3. Molecular generation. Matched scaffold (left: lower QVina, higher QED) and QM9 (right: lower property error) examples for each proposal. Tables 3 and 4 report performance. both VCG and RCPT preserve all 144 lineages; RCPT retains a mean amplitude of0.945and achieves slightly better mean evaluator error and unique success. These protocols demonstrate both repair of an unstable fit and retention of a well-supported update; Appendix Table I.6 reports seed variability. Table 4. Multi-property molecular steering. Eight-seed means. Unique success counts distinct valid target-meeting identities per 128 split or 144 online outputs. Lineages are normalized by batch size: N = 16 split, N = 144 online. ProtocolControllerEval. MAE↓Unique success↑Lineages/N ↑ Split 16 + 8× 16Anchor0.917120.75/1280.3311 VCG0.928113.88/1280.1611 RCPT0.899723.00/1280.3525 Online N = 144Anchor1.05700.75/1440.0069 VCG0.802745.25/1441.0000 RCPT0.799445.50/1441.0000 5.4 Improving Guided Image Generation on CIFAR-10 Using pretrained DLSM [7], we targetep t = q t p φ (c | x,t) 2.5 with matched Anchor guidanceγ = 2.5. VCG-9 and RCPT fit nine auxiliary class gradients on 16 states, then generate 320 images (32/class) with a frozen schedule and 500 probability-flow Euler steps. Independent ResNet accuracy and Inception, DINO, and CLIP metrics evaluate one seed (Appendix I.5). RCPT improves all six externally evaluated metrics over Anchor and VCG-9 (Table 5). Relative to VCG-9, independent accuracy rises from0.6438to0.6750, KID falls from0.2685to0.2407, and Clean-FID falls from89.759to88.232. A separate unit-power mechanism audit finds lower residual risk than VCG in 46/50, 47/50, and 43/50 class–seed cells at fitting populations 8, 16, and 32. Appendix Figure I.1 gives selected qualitative examples. Table 5. Class-conditional Image Generation on CIFAR-10. Single-seed comparison using all 320 outputs per method. KID uses independent ResNet features; FID and DINO FD use Inception and DINO features, respectively. DLSM accuracy measures guider self-consistency, not independent class fidelity. Controller Ind. acc.↑ DLSM acc.↑KID↓FID↓ DINO FD↓ DINO MMD↓ CLIP margin↑ Anchor0.66250.9625 0.2688 89.0430.247400.035390.00942 VCG0.64380.9813 0.2685 89.7590.248260.033810.00809 RCPT0.67500.96250.240788.2320.245340.032800.01001 6 Discussion Do existing regularization methods already solve the problem? The ablations separate changes to the fitted direction through ridge or PCR from changes to the deployed amplitude through direct or structured calibration. On common Checker fits atN = 16, RCPT’s mean Risk/Anchor is0.4931, versus0.8306for LOO-selected ridge and1.1740for PCR (Table 1); both paired 95% bootstrap intervals favor RCPT. RCPT captures80.7%of reference-ray improvement (Appendix Table B.1); Appendix B gives the paired intervals. Does reducing controller capacity solve the problem? Reducing capacity can stabilize residual fitting but sacrifice downstream chemistry. Across 76 scaffold pockets, RCPT improves validity over VCG-4 at both fitting budgets. At 8 fit 64, affinity-qualified yield rises from0.243to0.312and unique useful yield from0.206to0.271, with paired 95% intervals excluding zero, despite VCG-4’s lower residual risk (Appendix Table B.2; Appendix I.3). In CIFAR, RCPT has fewer harmful cells than VCG-4 (5/50 vs. 13/50), while VCG-4 has lower mean risk (Appendix Table B.6). Although oracle directionwise shrinkage can improve on a common amplitude, estimating additional amplitudes from finite particles can overfit deletion noise and increase deployment risk, motivating RCPT’s scalar calibration (Appendix F). The structured decision adds value beyond direct calibration. Against Direct LOO, RCPT improves paired Checker risk atN = 16and mean online QM9 unique success; its CIFAR advantages are independent accuracy, Clean-FID, and CLIP margin. On Scaffold, Direct LOO has lower common-state risk, while RCPT has stronger downstream means with paired intervals including zero (Appendix Tables B.3, B.4, and B.5). Do simple baselines already solve the problem? Fixed shrinkage, ridge, averaging, and norm caps can repair particular deployments, but their settings transfer unevenly. Fixedα = 0.50exceeds RCPT’s molecular yields on the ten-pocket scaffold subset, yet worsens mean QM9 evaluator error relative to VCG in the four-seed audit. Fixed ridge is similarly setting-sensitive. RCPT supplies one data-dependent rule across these tasks; the evidence supports adaptive protection, not uniform superiority over every tuned heuristic (Appendix Tables I.11 and I.12). Supported updates remain inexpensive. Figure 4 shows conservative scaffold and image amplitudes, time-varying retention on Checker, and strong retention on online QM9. In the scaffold capacity audit, increasing the fitting budget from 16 to 64 raises mean RCPT amplitude from0.194to0.906. Calibration adds no expert calls; measured QM9 overhead is3.13seconds, or0.081%, with equal probed peak memory. Appendix Table B.7 and Appendix K separate application cost from selector timings. 0.000.250.500.751.00 0.00 0.25 0.50 0.75 1.00 Structured RCPT amplitude α (a) Scaffold (n= 76) 0.000.250.500.751.00 (b) Reward-SMC (n= 8) 0.000.250.500.751.00 (c) Checker (n= 10) 0.000.250.500.751.00 (d) CIFAR-10 (n= 10) Normalized path time t Figure 4. RCPT adapts its trust across tasks. Mean amplitudes (thick), interquartile ranges (bands), and individual units (thin) against normalized native time. Zero recovers Anchor; one retains VCG. See Appendix J.2 for more details. Why not refit VCG online? Online refitting tracks changes in the particle distribution, but each update still estimates a controller from a finite ensemble. Withmparticles, the centered empirical feature covariance has rank at mostm− 1. Thus, repeated fitting does not by itself increase the independent information available at a given time. RCPT applies the same calibration principle to frozen and online fits. In the online QM9 experiment with 144 particles, it retains most of the VCG update, achieves slightly better terminal performance, and preserves all lineages (Table 4). This illustrates how calibration accommodates informative online fits while controlling the amplitude supported by the available particles. Limitations and future directions. The structured guarantee is conditional on calibration error, and residual upper bounds do not impose a universal ordering of endpoints or genealogy. The common-fitN = 32SMC audit favors Anchor over RCPT inW 2 by0.0817(paired 95% interval[0.0051, 0.1629]), while the larger scaffold deployment is essentially tied with Anchor. Extending frozen-schedule guarantees to online interacting particles requires dependence and target-reference controls; lineage-aware calibration and stable grouped shrinkage are natural next steps. Multiple fitting seeds per scaffold task and broader matched regularization cohorts would further test robustness. Appendix H.4 states the relevant transfer conditions. 7 Conclusion Finite-particle proposal fitting creates a deployment decision that population control alone does not resolve. RCPT selects how much of an estimated FK correction to apply using the existing regression information. The headroom– penalty identity explains the statistical failure, and realized-ray analysis makes the quality of the selected amplitude directly testable. Common-fit comparisons show a substantial advantage over adaptive regularization at a scarce-particle budget, while controlled and molecular deployments establish practical repair of harmful VCG fits. The resulting method provides an inexpensive way to retain supported adaptation when fitting information is limited. 9 AI Use Statement Large language models (LLMs) were used to revise author-written text for clarity, concision, and organization, and to assist with literature retrieval, theoretical formulation, mathematical derivations and proof writing, feedback on experimental design, code development, and interpretation of results. LLMs also assisted with drafting and revising manuscript passages and preparing figures, tables, references, and LaTeX. The authors determine the final research claims and take responsibility for the correctness, originality, and verification of all AI-assisted text, proofs, code, and interpretations included in this work. Ethics statement Generated molecules are computational candidates, not validated compounds. Docking and learned-property scores do not establish biological activity, synthesizability, or safety. The method inherits the limitations of the pretrained generators and evaluators used for steering. Reproducibility Statement Upon acceptance, we will provide supplementary source code for the structured RCPT estimator, the Direct LOO calibration ablation, and integrations for Checker, scaffold decoration, QM9 reward steering, and CIFAR-10. The package includes application drivers, configuration and dependency files, and CPU unit tests for the core estimator; its README describes setup commands, entrypoints, and external asset requirements. Appendices C–F state the theoretical assumptions and proofs. Appendix H specifies weighted fitting, deletion refits, and numerical guards, while Appendix I documents datasets, fitting and deployment protocols, evaluation metrics, and statistical units. Pretrained weights, datasets, generated samples, and experiment outputs are not bundled. These materials support inspection and testing of the calibration rule and document the setup required for application-scale reproduction. Acknowledgements This work was partly supported by the KHIDI grant funded by the Korean government (MOHW) [No.RS-2025- 02307233, No.RS-2026-25613012], the NRF or IITP grants funded by the Korean government (MSIT) [No.05-26-04- 0094, No.RS-2026-25472075, No.RS-2025-02305581, No.RS-2025-25442338, and No.RS-2021-I211343], the ITIP grant funded by the Korean government (MOTIR) [No.RS-2026-25549946], the Research grant from SNU, the Strategic Hub grant for International Research Collaboration of SNU, and the AI Seoul Tech Research Support Program of the Seoul Future Foundation. Kyungsu Kim is affiliated with the School of Transdisciplinary Innovations, Department of Biomedical Science, Interdisciplinary Program in AI (IPAI) & Graduate School of AI, Medical Research Center, and AI Institute at SNU. References [1]S. Agapiou, O. Papaspiliopoulos, D. Sanz-Alonso, and A. M. Stuart. Importance sampling: Intrinsic dimension and computational cost. Statistical Science, 32(3):405–431, 2017. 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In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pages 23174–23184, October 2023. 14 Appendix A Supplementary Overview16 B Additional Ablation and Cost Tables16 C Transport Residual and Correction Weights17 D Finite-Particle Risk and Structured Calibration21 E Calibration Ablation and ECG Baseline23 F Scalar and Directional Calibration23 G Specialization to FKC and ACE25 H Implementation Details25 IAdditional Evaluations27 J Additional Mechanism Diagnostics36 K Computational Cost38 15 A Supplementary Overview The supplement first presents additional ablations and cost measurements, then the proofs needed by the main paper, implementation details, and additional evaluations. RCPT denotes the structuredH,Prule in all reported tables and figures. Direct LOO is defined as a calibration ablation. Time runs from noise att = 0to the terminal law att = 1, unless an experiment states its native reverse-time coordinate. Scope of the guarantees.The transport bounds concern a fixed proposal schedule and the specified target path. The H,P,Bidentities average over fitting samples, while the structured calibration theorem controls a realized decision through its quadratic approximation errors. The directional oracle and counterexample explain the choice of one amplitude. None of these results orders all terminal metrics, certifies the online interacting-particle loop, or identifies weight ESS with independent regression information. Broader impact.RCPT may reduce the particle and memory cost of inference-time steering. It inherits the limitations and potential misuse of the underlying generators: generated molecules remain computational hypotheses, and image steering can also produce misleading content. Proposal calibration supplies no scientific validity or safety guarantee. B Additional Ablation and Cost Tables B.1 Common-fit calibration and paired comparisons The pairedN = 16comparison supports the structured decision. RCPT-minus-comparator Risk/Anchor differences are −0.6331for Direct LOO (95% CI[−1.363,−0.0669]),−0.3375for LOO-selected ridge (95% CI[−0.7484,−0.0510]), and−0.6804for LOO-selected PCR (95% CI[−1.182,−0.2518]). All three paired fitting-seed bootstrap intervals exclude zero, indicating lower mean reference risk at the pairwise 95% level, conditional on the reference construction. The reference-ray oracle quantifies the useful correction available along the fitted direction. For mean normalized risks ̄ R, define the descriptive fraction 1− ̄ R(M ) 1− ̄ R(ray oracle) . Anchor gives zero and the reference-ray oracle gives one. The diagnostic oracle risks are0.5758,0.3717, and0.2570at N = 8, 16, 32, respectively. Applying this expression to the reported aggregate risks gives RCPT approximately46.8%, 80.7%, and86.3%atN = 8, 16, 32. This is a ratio of aggregate improvements, rather than a mean of per-fit fractions. Table B.1 pairs it with the reported curvature-weighted ray regret. The oracle uses evaluation states and serves only as a diagnostic. Table B.1. Same-ray calibration ablation. Captured improvement uses the diagnostic oracle risks specified above and the mean normalized risks in Table 1. Regret is the mean integrated excess risk over the reference-ray oracle. All deployable rows use the same full-data VCG direction. N = 8N = 16N = 32 DecisionCapturedRegretCapturedRegretCapturedRegret Anchor0%15.7200%23.2900%27.540 Direct LOO17.9%12.900 −20.1%27.96083.2%4.635 RCPT46.8%8.35980.7%4.50086.3%3.776 B.2 Scaffold capacity and calibration The capacity study and matched calibration cohort answer different questions and retain separate means. Appendix I.3 gives the fitting and deployment protocols. 16 Table B.2. Scaffold controller-capacity ablation. The same 76 pockets and 16 deployment slots per arm. VCG-4 retains the ungated fields; VCG-12 and RCPT use all 12 fields. Mean RCPT amplitude is 0.194 at fit 16 and 0.906 at fit 64. All yields use the fixed generation denominator. Fit Controller Validity↑ Scaffold↑ Risk/Anchor↓ Dock success↑ Affinity↑ Unique useful↑ 16VCG-40.3830.3740.6500.8670.2570.225 16VCG-120.2290.22526.2750.5690.1570.118 16RCPT0.4340.4190.8770.9030.2970.251 64VCG-40.3540.3350.2970.9360.2430.206 64VCG-120.3350.3221.0690.9150.2360.200 64RCPT0.4410.4220.5480.9110.3120.271 Table B.3. Matched Scaffold calibration ablation. A separate fit-16/deploy-16 cohort on all 76 pockets. Anchor, Direct LOO, and RCPT isolate the addition of calibration and its structured form; the full VCG fit is shown for reference. These means are kept separate from the earlier cohort. Risk is the active-path diagnostic; paired uncertainty is reported in Appendix I.3. ControllerValidity↑ Scaffold↑ Risk/Anchor↓ Dock success↑ Affinity↑ Unique useful↑ Anchor0.4380.4151.0000.8980.2940.251 VCG-120.2200.21321.1840.5620.1430.111 Direct LOO0.4180.4000.7560.9140.2890.247 RCPT0.4380.4230.8760.9060.3030.262 B.3 QM9 and image calibration These ablations use the online QM9 budget and the CIFAR terminal-generation cohort described in Appendices I.4 and I.5. Table B.4. QM9 calibration ablation. Eight online seeds,N = 144. Anchor has no fitted correction; Direct LOO uses direct deleted-loss calibration; RCPT uses the structured headroom–penalty decision. Unique success counts distinct valid target-meeting identities per 144 outputs. CalibrationMAE↓Unique success /144↑Lineages/N ↑ Anchor1.05700.750.0069 Direct LOO0.801943.130.9479 RCPT0.799445.501.0000 Table B.5. CIFAR-10 calibration ablation. The same 320-output seed and nine auxiliary fields as Table 5. Direct LOO and RCPT calibrate the same fitted direction. Bold marks the best value in each column. ControllerInd. acc.↑ DLSM acc.↑KID↓FID↓ DINO FD↓ DINO MMD↓ CLIP margin↑ Anchor0.66250.9625 0.2688 89.0430.247400.035390.00942 Direct LOO0.65630.9656 0.2308 88.7220.245230.032410.00979 RCPT0.67500.96250.240788.2320.245340.032800.01001 B.4 Application cost In the common-fit Checker implementation, mean selector times per 200-step schedule are0.132seconds for RCPT,1.31 for Direct LOO,9.34for selected ridge, and13.03for selected PCR. These implementation-specific selector timings are separate from the hardware-identified generation measurements below. Appendix K gives the instrumentation. C Transport Residual and Correction Weights 17 Table B.6. CIFAR-10 target-matched risk and capacity audit. Fifty class–seed cells, five seeds, fit 16, andβ = γ = 2.5. Each independent-state risk is divided by its cell’s matched Anchor risk before averaging. These are common-state surrogate risks. Harm means a ratio above one; bold marks the best fitted method. ControllerMean Risk/Anchor↓Harm↓Worst ratio↓ Anchor1.00000/501.0000 VCG-90.996822/501.0246 VCG-40.986913/501.0401 Direct LOO0.997013/501.0204 RCPT0.99775/501.0126 Table B.7. Measured application cost. Scaffold means use all 76 pockets on an RTX A6000 (48 GB). QM9 uses four recorded A100-SXM4 (80 GB) seeds; peak memory is from a matched 10-step probe. Times exclude offline evaluation. Full instrumentation is in Appendix K. SettingControllerGeneration time (s)Peak memory (GB) Scaffold, 16 particlesVCG201.7155.070 RCPT197.4574.742 QM9, 144 particlesVCG3879.51840.204 RCPT3882.64540.204 C.1 Analytic setup Letep t > 0have finite normalizerZ t and sufficient space-time regularity to differentiate under the integral. For each fixed admissible drift schedule, assume the conservative operatorL ∗ t,v generates a Markov evolutionU v,∗ s,t , the forward equation has a unique mild solution, and R 1 0 ∥p t r v t ∥ L 1 dt <∞. Boundary fluxes vanish. Assumer v t ∈ L 2 (p t ) for the stated risk bounds. Random schedules fitted independently of deployment can be conditioned on throughout. These assumptions do not identify a particle-dependent online drift with a fixed single-particle Markov process. Nonsmooth or vanishing terminal densities require a limiting argument. One may apply the dynamic result up to1− τ on a regular positive path, then letτ ↓ 0if the endpoint laws converge inW 2 and the energy bound remains uniformly integrable. The triangle inequality and this convergence pass the upper bound to the limit. C.2 Proof of Theorem 4 Proof.The definition ofg v t gives∂ t ep t =L ∗ t,v ep t + g v t ep t . Integrating and using conservativity yields ̇ Z t /Z t = E p t g v t . Substituting ep t = Z t p t proves ∂ t p t =L ∗ t,v p t + p t r v t , r v t = g v t − E p t g v t .(C.1) Since p 0 = ρ v 0 , variation of constants gives p 1 − ρ v 1 = Z 1 0 U v,∗ t,1 (p t r v t ) dt.(C.2) Markov kernels contract the total-variation norm of signed measures. Therefore, with d TV (p,q) = 1 2 ∥p− q∥ 1 , d TV (p 1 ,ρ v 1 )≤ 1 2 Z 1 0 E p t |r v t | dt≤ 1 2 Z 1 0 p V t (v) dt. This proves the normalized source identity. The Wasserstein part follows in Appendix C.3. For completeness, the optional final time relaxation is d TV (p 1 ,ρ v 1 )≤ 1 2 Z 1 0 p V t (v) dt≤ 1 2 s Z 1 0 V t (v) dt.(C.3) 18 For a statisticm ∈ [a,b], the same argument implies|E p 1 m− E ρ v 1 m| ≤ (b− a)d TV (p 1 ,ρ v 1 ). Zero residual along the entire path implies exact unweighted marginals by uniqueness. A reduced residual bound alone does not imply exactness or a strict reduction in terminal discrepancy: the propagated signed defects in Eq. (C.2) can cancel. C.3 Proof of the Wasserstein upper bound Transport energy retains spatial information that residual variance discards. For a zero-mass signed density h, define ∥h∥ −1,p =sup R ∥∇f∥ 2 p≤1 Z fh , A p =−∇· (p∇).(C.4) We use weak solutions modulo constants, with zero boundary flux or sufficient decay at infinity. IfA p ψ = hhas finite energy, testing its weak equation and using Cauchy–Schwarz gives∥h∥ 2 −1,p = R ∥∇ψ∥ 2 p =⟨h,A −1 p h⟩ . In particular, E t (v) =∥p t r v t ∥ 2 −1,p t . Lemma C.1 (Minimum correction energy). Among velocities satisfying−∇· (p t w) = p t r v t , the gradientw =∇ψ t minimizes E p t ∥w∥ 2 . If p t has Poincaré constant C P (t), thenE t (v)≤ C P (t)V t (v). Proof. Every admissiblewdiffers from∇ψ t by a fieldzwith∇· (p t z) = 0. Integration by parts givesE p t ∇ψ t ·z = 0, proving minimality by the Pythagorean identity. Choose the mean-zero representative of ψ t . Then E t (v) = E p t (r v t ψ t )≤ p V t (v)C P (t)E t (v). Division proves the claim when the energy is positive; the zero case is immediate. Proof of the Wasserstein bound in Theorem 4. The corrected driftv t +∇ψ t has marginalsp t by Eq. (C.1) and uniqueness. Couple its processX t with the proposalY t using the same additive Brownian noise and an optimal initial coupling. The noise cancels. With D t = (E∥X t − Y t ∥ 2 ) 1/2 , the one-sided Lipschitz condition implies 1 2 ∂ t D 2 t ≤ L t D 2 t + D t p E t (v). Regularization at D t = 0 and Gronwall give the slightly more general bound W 2 (p 1 ,ρ v 1 )≤ K v (0, 1)W 2 (p 0 ,ρ v 0 ) + Z 1 0 K v (t, 1) p E t (v) dt.(C.5) This proves Eq. (14). Poincaré is needed only to replace energy by variance. The energy bound itself does not require it. What correction changes.Exact FK weighting targetsp t despite proposal mismatch. The preceding terminal bounds concern the unweighted proposal lawρ v t . They do not bound the error of a finite corrected particle system. The local weight result below explains a separate benefit of reducing residual variance. Particle error additionally depends on weights, dependence, resampling, and mixing [11]. C.4 Proof of Proposition 1 Proof. WriteW − 1 = ∆r v t (X t ) + R ∆ with∥R ∆ ∥ 2 = o(∆). Then E(W − 1) 2 = ∆ 2 E p t (r v t ) 2 + 2∆E[r v t (X t )R ∆ ] + ER 2 ∆ = ∆ 2 V t (v) + o(∆ 2 ). Because EW = 1, this equals both EW 2 − 1 and χ 2 (P∥Q). One sufficient route to the assumed expansion isA ∆ = R t+∆ t g v s (X s ) ds = ∆g v t (X t ) + o L 2 (∆)together with ∥e A ∆ − 1−A ∆ ∥ 2 = o(∆). Normalizinge A ∆ then subtractsE p t g v t at first order. For example, bounded rates with local L 2 continuity satisfy these conditions. Uniform conclusions require uniform remainder control. 19 C.5 Why local variance does not order genealogy LetZbe uniform on−1, 1and0 < ε < 1. Two weighting schemes useG 1 = 1 + εZand respectivelyG 2 = 1− εZ orG 2 = 1 + εZ. Every increment has mean one and squared coefficient of variationε 2 under this common reference law. However, CV 2 (G 1 G 2 ) = 0 or 4ε 2 (1 + ε 2 ) 2 , respectively. Thus marginal incremental second moments do not determine cumulative weight variation. Resampling adds further dependence on timing. For exactly Gaussian log weight L with variance s 2 , Ee 2L (Ee L ) 2 = e s 2 ,ESS pop /N = e −s 2 ,(C.6) but the variance of a sum of log increments includes their cross-covariances. For multinomial resampling with weightsw i , the probability that all children share one parent is P i w N i . Resampling resets numerical weights without restoring ancestors. We therefore report both the number of surviving initial ancestors and lineage ESS: N anc =|a : π a > 0|,ESS anc = ( X a π 2 a ) −1 ,(C.7) where π a is the fraction of descendants of ancestor a. C.6 Time discretization and approximate residuals On a grid 0 = t 0 <· < t K = 1, hold λ k fixed on each interval. If|∂ t V t (λ k )|≤ L k , then Z t k+1 t k V t (λ k ) dt− ∆ k V t k (λ k ) ≤ 1 2 L k ∆ 2 k .(C.8) Consequently the continuous-time proposal with that schedule satisfies d TV (p 1 ,ρ λ 0:K−1 1 )≤ 1 2 s X k ∆ k V t k (λ k ) + 1 2 X k L k ∆ 2 k .(C.9) The discretized sampler has an additional errord TV (ρ λ 1 ,ρ λ 1,h ); the bound does not assert a universal rate for it. If r t = br t + δ t , then ∥r∥ L 2 ( dtp t ) ≤∥br∥ L 2 ( dtp t ) +∥δ∥ L 2 ( dtp t ) .(C.10) Score, divergence, and tracked-density errors therefore remain separate from coefficient-estimation error. Wasserstein model and numerical errors.The direct transport argument also separates approximation errors without converting them to TV. If the true centered source is r t = br t + δr t , then ∥p t r t ∥ −1,p t ≤∥p t br t ∥ −1,p t +∥p t δr t ∥ −1,p t ≤ q b E t + p C P (t)∥δr t ∥ L 2 (p t ) . The last step alone uses Poincaré. A drift errorδv t changes the source by a weighted divergence, whose negative norm is at most∥δv t ∥ L 2 (p t ) by the dual definition. Apply Eq. (C.5) to the actual drift with its corresponding stability factor. These are deterministic model-error terms, not extra coefficient-estimation penalties. For a piecewise schedule, letF (t) = K v (t, 1) p E t (v)and suppose|F (t)− F (t k )| ≤ ω k (t− t k )on itskth interval. Then W 2 (p 1 ,ρ v 1 )≤ X k ∆ k F (t k ) + X k Z ∆ k 0 ω k (s) ds. This follows by integrating the modulus bound and preserves the energy profile until the final quadrature. A discretized lawρ v 1,h and an empirical outputbρ v 1,h add, by the triangle inequality,W 2 (ρ v 1 ,ρ v 1,h )andW 2 (ρ v 1,h ,bρ v 1,h ). Finite reference samples add their own error. For SMC, this last term is an interacting-particle error and cannot be replaced by an iid sampling rate without justification. 20 D Finite-Particle Risk and Structured Calibration D.1 Population projection The linear residual formula follows fromL ∗ v+u p = L ∗ v p−∇· (pu) . Expanding its variance givesV (λ) = V (0) + 2d ⊤ λ + λ ⊤ Cλ. For anyz ∈ kerC,z ⊤ (a− Ea) = 0almost surely, soz ⊤ d = 0. Thusd∈ range(C)and completing the square proves Eq. (3), including singular C. This is the population VCG projection of Ren et al. [33]. D.2 Proof of Theorem 1 Proof.Apply Eq. (3) to each realized fit and take expectation:R( b λ) = V ⋆ + P. At Anchor,V (0) = V ⋆ + H. Subtraction proves the identity. D.3 Proof of Theorem 2 Proof. Set e = b λ− λ ⋆ . Then R(α b λ)− V ⋆ = E∥− (1− α)λ ⋆ + αe∥ 2 C = (1− α) 2 H + α 2 P − 2α(1− α)B. The coefficient ofα 2 isD 0 = H + P + 2B = E∥ b λ∥ 2 C ≥ 0 . ForD 0 > 0, minimizing the convex quadratic gives Eq. (6); clipping enforces the interval constraint. WhenB = 0, substitution ofH/(H + P )gives Eq. (7). IfD 0 = 0, the fitted correction is constant p-almost surely for almost every fit, so its amplitude has no effect on variance. D.4 Proof of Theorem 3 Proof. SinceS, b P ≥ 0andε num > 0, the minimizer of Eq. (11) isΠ [0,1] ((S− b P )/(S + ε num )). It equals Eq. (10), including S ≤ b P . For the realized field, write ∆(α) = V (α b λ)− V (0) = qα 2 − 2(q− κ)α.(D.1) Forq > 0, the true minimizer isα ray = Π [0,1] (1− κ/q); forq = 0, the risk is constant. Lete = e q + 2e κ . Since |α 2 − 2α|≤ 1 on [0, 1], −e≤ b ∆ S (α)− ∆(α)− ε num α 2 ≤ e. Optimality and b ∆ S (0) = 0 imply ∆(bα)≤ b ∆ S (bα) + e≤ e. Comparing instead to α ray gives ∆(bα)≤ b ∆ S (α ray ) + e≤ ∆(α ray ) + 2e + ε num . Finally, for a fixed dictionary, d =−Cλ ⋆ gives κ = b λ ⊤ C( b λ− λ ⋆ ). Taking expectation yields E D κ = P + B. If the estimation-error event has probability at least1− δ, the same conclusions hold with that probability. Establishing such an event requires control of the sampling law, fitted field, and residual estimator. The algebraic bound does not supply this control automatically. D.5 What the structured model assumes Bias sensitivity of the deterministic oracle.Writeα 0 = H/(H + P )forH + P > 0. For nonzeroB, letα B be the constrained oracle in Eq. (6). Then 0≤R(α 0 b λ)−R(α B b λ)≤|B|/2.(D.2) Indeed, the omitted term−2Bα(1− α)has range width|B|/2, andα 0 minimizes the remaining quadratic. When the bias-aware oracle is interior and D 0 = H + P + 2B > 0, the exact regret is B 2 (P − H) 2 (H + P ) 2 D 0 , H + B D 0 − H H + P = B(P − H) (H + P )D 0 . These statements concern population constants and deterministic amplitudes. They do not transfer unchanged to an estimatedbα. In particular,E∥ b λ∥ 2 C = H + P + 2B, so subtracting a penalty estimate does not universally give an unbiased estimate of H . 21 An exact normal-means interpretation. In whitened coordinates, suppose Z | θ ∼N (θ,σ 2 I r ), θ ∼N (0,τ 2 I r ). Completing the square givesE[θ | Z] = τ 2 (τ 2 + σ 2 ) −1 Z. This posterior mean minimizes posterior squared loss. With H prior = rτ 2 andP = rσ 2 , its amplitude is exactlyH prior /(H prior + P ). For knownσ 2 , maximizing the marginal normal likelihood givesbτ 2 = [∥Z∥ 2 /r−σ 2 ] + , whose plug-in amplitude is[1−rσ 2 /∥Z∥ 2 ] + (zero whenZ = 0). This is the RCPT algebra withS = ∥Z∥ 2 , known penaltyP, andε num = 0. It is a classical empirical-Bayes shrinkage interpretation [15], not a claim that the fitted diffusion controllers follow this model. Estimating the hyperparameters, geometry, and penalty does not inherit exact posterior optimality. Why Direct LOO need not improve the decision.For an interior realized oracle,∆(α)− ∆(α ray ) = q(α− α ray ) 2 . A lower-variance estimate of the amplitude can therefore be preferable even if its criterion is biased. At a boundary, the exact regret also includes2(qα ray − (q− κ))(α− α ray )≥ 0. The criterion-estimation tradeoff is consistent with established comparisons of covariance penalties and cross-validation [14]. It is not a universal dominance result for structured RCPT. The common-fit audit in Appendix I.2 illustrates the distinction: smaller unweighted amplitude error need not give smaller curvature-weighted ray regret. Accurate quadratic coefficients are sufficient, but not necessary, for a good decision; multiplying the entire criterion by a positive constant leaves its minimizer unchanged. Likewise, a zero-mean criterion correction at each fixed α need not have zero mean after minimizing over α. D.6 Finite-particle penalty and its proof Lemma D.1 (Finite-particle estimation penalty). LetX 1 ,...,X N iid ∼ p with a fixed dictionary,A = a − E p a, ε = g 0 − E p g 0 + A ⊤ λ ⋆ , andΩ = E p [ε 2 A ⊤ ]. SupposeC = E p A ⊤ has fixed rank, its nonzero eigenvalues are positive,E∥A∥ 4 < ∞,Eε 2 < ∞,E[ε 2 ∥A∥ 2 ] < ∞, and empirical rank is eventually equal to population rank in probability. Use minimum-norm least squares, or ridge η N = o(N −1/2 ) in the normalized objective. On range(C), √ N ( b λ− λ ⋆ ) d −→N (0,C † ΩC † ).(D.3) IfN∥ b λ− λ ⋆ ∥ 2 C is also uniformly integrable, then P = N −1 tr(ΩC † ) + o(N −1 ). Proof. The population normal equation isEAε = 0. The centered empirical normal equations and consistency of the covariance inverse on its range give b λ− λ ⋆ =−C † 1 N X n A n ε n + o p (N −1/2 ).(D.4) Sample centering contributesO p (N −1 ); the stipulated ridge contributeso p (N −1/2 ). Apply the multivariate CLT. The quadratic continuous-mapping theorem and uniform integrability then yield NP → tr(C † ΩC † ) = tr(ΩC † ). The reduction toσ 2 ε rank(C)/N requiresΩ = σ 2 ε C . RCPT further substitutes a deleted prediction residual forσ 2 ε , effective rank for rank, and weight ESS for sample count. These choices define its low-particle model; the asymptotic lemma does not assert that they are accurate near interpolation. Ridge and dependent particles. Fixed population ridge targets λ η =−(C + ηI) −1 d and incurs V (λ η )− V (λ ⋆ ) = η 2 (λ ⋆ ) ⊤ (C + ηI) −1 C(C + ηI) −1 λ ⋆ .(D.5) Thus the preceding centered CLT is not a fixed-ridge theorem. For independent observations with deterministic normalized weights, analogous weighted normal equations give leading variance( P n ω 2 n ) tr(ΩC † ) under weighted consistency, a weighted CLT, and uniform integrability. Random importance weights and interacting particles require their own covariance analysis. If a particle-system CLT for the regression score has covarianceΓ t , the same linearization, covariance consistency, and uniform integrability giveN −1 tr(Γ t C † )instead. Conventional weight ESS does not estimate this covariance in general; equal weights after resampling do not imply independent observations. 22 Frozen-schedule consequence. For a schedule fitted independently of deployment, combine the projection identity with the dynamic Wasserstein bound. If e λ k is held on [t k ,t k+1 ) and the assumptions of Theorem 4 hold, then W 2 (p 1 ,ρ e λ 1 )≤ X k Z t k+1 t k K v (t, 1) q C P (t)V t,⋆ +∥ e λ k − λ ⋆ t ∥ 2 C t dt.(D.6) This retains the time profile of the estimation error. Theorem 3 controls the selected coefficient at each grid point under its calibration assumptions. Interpolation, model error, and numerical integration require the separate controls above. A smaller upper bound alone does not order endpoint errors. A guarantee for the same-particle online loop additionally requires analysis of that interacting process. E Calibration Ablation and ECG Baseline E.1 Direct LOO Using the deleted means and fits in the main text, setu n = g 0 n − ̄g 0 −n andz n = (a n − ̄a −n ) ⊤ b λ (−n) . Define A D = P n ω n u 2 n , b D = P n ω n u n z n , and c D = P n ω n z 2 n . Then b L D (α) = A D + 2b D α + c D α 2 , bα D = ( Π [0,1] (−b D /c D ), c D > 0, 0,c D = 0. (E.1) The formula follows by minimizing a scalar quadratic. The intercept must be ̄g 0 −n + α ̄a ⊤ −n b λ (−n) at each amplitude; holding the unscaled intercept fixed changes the objective. Both selectors deploy the full-data coefficient, not an average of deleted coefficients. Deletion weights are renormalized and the same ridge normalization is used in every fold. Direct LOO estimates deleted prediction loss without a scalar covariance model. It changes only the calibration criterion, while retaining the full-data VCG direction. Appendix H.4 distinguishes target-law risk from the reference- law diagnostics and numerical guards used in deployment. E.2 Residual fitting versus energy fitting VCG minimizesVar p (g 0 +T p u)in a vector-field class. For gradient controlsu =∇ψ, energy-controlling guidance (ECG) instead uses the weak Poisson objective [33] min ψ 1 2 E p ∥∇ψ∥ 2 − E p [r 0 ψ].(E.2) Its first variation isE p ∇ψ·∇h = E p r 0 h, equivalent under integration by parts to−∇·(p∇ψ) = pr 0 . Forψ = P j θ j ψ j , ECG solvesMθ = b, withM ij = E p ∇ψ i ·∇ψ j andb i = Cov p (ψ i ,g 0 ). The two finite-dimensional objectives need not select the same field. ECG provides a distinct proposal-fitting baseline. Its scalar-potential class does not automatically contain arbitrary gated vector fields used by VCG and RCPT. Conversely, ECG’s energy objective avoids squared Stein features in its normal matrix. These are representation and objective differences, not a general superiority result. ECG also estimates its moments from finite particles. Comparisons should state the available field class, fitting budget, and whether correction weights and resampling are retained; no resampling alone does not mean unweighted Proposal-Only sampling. F Scalar and Directional Calibration F.1 The oracle benefit of directional shrinkage Scalar shrinkage is a restricted decision class, not an isotropy assumption. On the range ofC, writeθ = C 1/2 λ ⋆ and Z = C 1/2 b λ = θ + ξ. Suppose Eξ = 0 and define H j = θ 2 j , P j = Eξ 2 j . For fixed diagonal amplitudes, E∥ diag(α j )Z− θ∥ 2 = X j (1− α j ) 2 H j + α 2 j P j .(F.1) No independence between the coordinates is needed. Cross-covariances do not enter a sum of coordinatewise squares. 23 Proposition F.1 (Exact directional oracle gain). WriteT j = H j + P j . If allT j = 0, every amplitude has the same risk. Otherwise omit coordinates withT j = 0. The directional oracle isα ∗ j = H j /T j , the scalar oracle is α ∗ s = P j H j / P j T j , and their risk difference is R ∗ scalar −R ∗ directional = X j T j (α ∗ j − α ∗ s ) 2 ≥ 0.(F.2) Proof.Complete each square in Eq. (F.1). The unconstrained minimizers lie in[0, 1]. Under a common amplitude, minimizing the sum gives α ∗ s . Subtract the coordinatewise minima to obtain the identity. For example, letθ = (1, 0)andξ = (0,ε)withEε = 0,Eε 2 = L > 0. The directional oracle keeps the first coordinate and removes the second, giving zero coefficient risk. The scalar oracle has riskL/(1 + L). Grouping coordinates replacesH j ,P j by their group sums. A zero-amplitude group is a controller truncation. Thus reduced rank and shrinkage are complementary decisions. These are oracle comparisons; estimating additional amplitudes has a cost. F.2 An exact sample on which directional LOO is worse Extra amplitudes can fit deletion noise even when they use the same fits. Letabe uniform one 1 ,−e 1 ,e 2 ,−e 2 , letg be an independent uniform sign, and use η = 1, N = 4. The ordered sample D = ((e 1 ,−1), (e 1 ,−1), (e 2 , 1), (−e 2 , 1)) has probability 8 −4 . Direct calculation gives b λ D = (2/5, 0) and the deleted residual representation u i + v ⊤ i α with u = (−4/3,−4/3, 4/3, 4/3) ⊤ ,(v ⊤ i ) 4 i=1 = 8/330 8/330 −8/39 16/39 −8/39 16/39 .(F.3) The first two deletion coefficients are(4/11, 0); the last two are(4/13, 4/13)and(4/13,−4/13), which verifies the matrix using the deleted means. The scalar criterion has minimizer143/290. For the two-coordinate criterion, at(1, 0) both residual types are nonzero, the derivative in coordinate one is negative, and the derivative in coordinate two is positive. These are the constrained optimality conditions at that corner. The two distinct row vectors span the plane, so strict convexity makes (1, 0) the unique minimizer. Population risk is V (λ) = 1 +∥λ∥ 2 /2. Deploying the full fit with these amplitudes therefore gives V scalar = 1 + 2 25 143 290 2 ≃ 1.01945, V directional = 1 + 2 25 = 1.08. This is a positive-probability counterexample to samplewise dominance of directional LOO. It is not a comparison of their risks averaged over all possible fitting samples. Together with Proposition F.1, it separates the benefit of a richer oracle from the reliability of estimating it. F.3 When the bias term can be omitted Projection orthogonality alone does not implyB = 0. One sufficient condition is a correctly specified conditional linear modelg = c−a ⊤ λ ⋆ +ε, withE[ε| a] = 0, full empirical rank, unregularized least squares, and integrable coefficients. Conditioning on the design gives E[ b λ| a 1:N ] = λ ⋆ , so B = 0. This is stronger than the population normal equation. A centered first-order CLT is also insufficient by itself. A coefficient bias of orderN −1 can makeBthe same order as the estimation penaltyP. The general bound|B| ≤ √ H∥E b λ− λ ⋆ ∥ C ≤ √ HPfollows from Cauchy–Schwarz, but need not be small relative toP. In a noiseless scalar example with ridge contraction b λ = cλ ⋆ ,0 < c < 1, one hasP = (1− c) 2 HandB = −(1− c)H. The constrained risk oracle retains the entire fitted update, whereas H/(H + P ) < 1shrinks it again. This proves that ridge can invalidate the zero-bias interpretation even without sampling noise. The paper’s bias audit is empirical evidence for its audited regime, not a universal justification for dropping B. 24 G Specialization to FKC and ACE Let expertiprovide densityq (i) t , scores (i) t = ∇ logq (i) t , and probability-flow velocityv (i) t . For a valid exponent schedule γ i (t), p t (x) = 1 Z t I Y i=1 q (i) t (x) γ i (t) , s t (x) = X i γ i (t)s (i) t (x).(G.1) In the implementation’s SDE parameterization, the selected probability-flow component enters g v t =∇· v t + v t · s t + b t ,(G.2) up to a spatial constant, where η t is the scalar linear-drift coefficient fixed by the underlying diffusion schedule and b t (x) = η t x· s t (x) + X i ̇γ i (t) logq (i) t (x) − X i γ i (t) n ∇· v (i) t (x) + v (i) t (x)· s (i) t (x) o .(G.3) The omitted divergence∇· (η t x) = dη t is spatially constant and irrelevant to normalized weights. The generic theory requires a valid path. VCG applies directly when the fixed-exponent path is valid. If fixed exponents create an invalid or singular intermediate path, ACE first repairs the schedule while preserving the desired endpoints [26]. Proposal fitting then acts on the repaired path. RCPT does not substitute for path repair and does not change Eq. (G.1). H Implementation Details H.1 Stable weighted least squares For normalized current particle weights ω n , the code minimizes min λ,c N X n=1 ω n g 0 n + A n λ− c 2 + η∥λ∥ 2 2 .(H.1) The intercept is eliminated by weighted centering. Covariances and them× mlinear solve are computed in float64; singular systems use a pseudoinverse. A coefficient clip is available in the high-dimensional experiments. After clipping, the code recomputes the selected empirical variance and uses the anchor coefficientλ = 0if it increased beyond a floating-point tolerance. The chosen proposal is used consistently for state propagation and all tracked logq i updates. Theorem 3 analyzes the scalar-selected field before an additional coefficient clip. If clipping changes it toλ final , let e clip =∥λ final − bα b λ∥ C . The triangle inequality for centered residuals gives p V (λ final )≤ q V (bα b λ) + e clip .(H.2) An Anchor fallback has exactly the Anchor risk, but an empirical-variance guard alone is not an out-of-fit guarantee. The realized deployed field and any clipping therefore need to be distinguished in applying the theory. The logged amplitude is the scalar selected before componentwise clipping; if an empirical guard falls back to Anchor, it remains a diagnostic of the pre-guard decision rather than the final coefficient. The logged diagnostics are the anchor variance, selected variance, regularized within-class floor, removable fraction R 2 t = d Var(g 0 t )− b V t,∗ d Var(g 0 t ) ,(H.3) coefficient norm, conditional ESS before and after selection, conventional ESS, and resampling events. Same-particle floors are optimization diagnostics, not held-out estimates of the population floor. 25 H.2 Basis and divergence by domain Two dimensions. The controlled ACE experiment anchors at its standard velocity and uses every coordinate vector field whose scalar coefficient is a monomial of total degree at most two. Exact automatic differentiation computes divergence in two dimensions. Images. We use the official pretrained DLSM CIFAR-10 score model and noise-conditioned classifier at upstream commit259c767. For classc, writeh c = logp φ (c| x,t),u c =∇h c , andb t =∇ logq t . For targetep t = q t exp(βh c ) and matched classifier-guided VE probability-flow Anchor withγ = β, the raw response in generation times = 1−tis g PF β,β =−β∂ t h c + βa(t)b ⊤ t u c + 1 2 a(t) β∆h c + β 2 ∥u c ∥ 2 2 ,(H.4) wherea(t)is the forward variance rate. This identity assumes an exact VE heat-flow base density; learned scores and Hutchinson traces yield a plug-in approximation. The response is centered for residual fitting. For the terminal cohort, β = γ = 2.5. The implementation represents this Anchor as a scale-1.5 base plus the same fixed+1target-class coefficient in every arm. Only the nine remaining fieldsφ j = 1 2 a(t)∇ logp φ (j | x,t),j ̸= c, are fitted, with target Stein featuresT p t φ j =∇· φ j + φ ⊤ j (b t + βu c ). Thus zero auxiliary correction recovers the matched scale-2.5 Anchor. One realized Rademacher Hutchinson probe is shared across methods within each class–seed–time cell. Fit states are forward-noised training images; independent test images provide held-out states. Ridge is10 −4 . Schedules are fitted at 16 midpoint times and linearly interpolated during 500-step deterministic VE probability-flow Euler sampling (PFO-500). The separate unit-power mechanism audit usesβ = γ = 1. Neither terminal images nor evaluation metrics enter fitting or calibration. Forward-noised labeled data are a proxy for the model-defined target path; these common-state risks do not directly evaluate its exact expectation. Molecules.States are flattened padded molecular coordinates and attributes, with all existing projections and masks retained. In scaffold decoration, each of the four expert probability-flow velocities is crossed with degree zero, one, and two bounded relative-density Legendre gates, giving the fixed 12-field basis in Eq. (I.2). Sixteen particles fit the VCG or RCPT schedule, and that schedule is frozen for independent deployment. Finite controller coefficients, feature matrices, state increments, and residuals are checked before every accepted update; these VCG-based arms share the stated numerical guards. Reward-tilted molecules. The EEGSDE basis [3] begins withν t ∇R polarizability,t andν t ∇R gap,t , whereν t is the one-step reverse-transition variance, and crosses these families with the fixed bounded Legendre gates. Each resulting field is normalized by its fit-batch RMS over active state coordinates before regression; the frozen schedule stores these scales. The base conditional diffusion drift and scaled FKC Anchor are fixed parts of the proposal and are not counted as controller fields. Autodifferentiation supplies reward gradients and Stein divergences. A same-state time finite difference evaluates both ̇ β j,t R j,t andβ j,t ∂ t R j,t in the reward-path potential. Generation uses the training reward networks; all reported properties use independently trained evaluators, and molecular identities use RDKit canonical isomeric SMILES. H.3 Deletion scope and computational complexity Deletion refits in this description act on the evaluated regression features. When gate statistics or field normalizations are estimated from the same cohort and then held fixed across folds, this is feature-level deletion rather than deletion of the entire field-construction pipeline. The fixed-dictionary statistical interpretation requires that construction to be fixed or conditioned on with a justified sampling law. The realized-field bound in Theorem 3 still applies, but its calibration errors must include any resulting adaptivity. No iid claim is made solely because the feature matrix has been frozen. After field values and divergences are available, weighted covariance formation and the VCG solve costO(Nm 2 +m 3 ). RCPT reuses those features and adds only dense leave-one-out algebra, with no expert or denoiser call. Hardware- identified timing, peak VRAM, and measured particle scaling are reported in Appendix K. 26 Algorithm H.1 Online VCG/RCPT Sampling Loop Input: valid path, Anchor, basis, ridge, correction flag, particlesX [n] 0 ∼ p 0 . Initialize log weights and tracked expert log densities. At each step: 1. Evaluate target score, Anchor, fields, and divergences; form the rate and feature matrix. 2. Normalize weights and solve Eq. (H.1). For RCPT, compute deletion residuals and the structured amplitude. 3. Apply the configured coefficient clip and empirical-variance fallback. 4. Propagate under the selected drift and update tracked densities using that same drift. 5. If correction is retained, update FK weights. If the resampling rule triggers, resample states and tracked quantities with common ancestry and reset weights. Return particles, weights (if retained), and diagnostics. H.4 Evaluation laws and dependent particles The theoretical risk is defined under the target marginalp t . Common-state evaluation compares methods on identical states but estimates risk under the reference law; active-path evaluation also changes the state distribution with the proposal. Neither is automatically a target-law expectation. The Checker corrected-SMC reference is an approximation, and the molecular common-state audit does not quantify its gap to the target law. Equal weights after resampling do not imply independent fitting information. Particles can share ancestors, and deleting one descendant can leave closely related states in the fitting ensemble. Weight ESS therefore does not by itself quantify regression information or justify an online guarantee. Lineage-aware deletion and particle-system concentration require a separate analysis [11]. Data-derived features, coefficient clipping, and empirical fallbacks are part of the deployed estimator. The conditional structured bound evaluates a fixed fitted direction; additional transformations need their own error control, and an empirical Anchor fallback is not an independent out-of-fit certificate. These distinctions delimit the interpretation of the reported risk audits. I Additional Evaluations I.1 Controlled targets Checker. Draw U ∼ Unif[−2, 2], V ∼ Unif[0, 1], and K ∼ Bernoulli(1/2), set X raw = 2U, Y = 2V − 2K + (⌊U⌋ mod 2), X = (X raw + 2 mod 8)− 4,(I.1) and condition onA = 1X ≥ 0 = 1andB = 1X + Y ≥ 0 = 1. This gives the exact alternating checker law on [−4, 4] 2 . Three separately trained expert paths define the ACE target. Anchor is only the learned joint expert velocity v XY|B ; it is not the sum of all three expert velocities. The other paths enter the time-dependent multiplicative target, while VCG and RCPT add a 12-field coordinate-polynomial correction (six monomials of total degree at most two for each of two velocity coordinates) to the shared Anchor. At every fitted time, Anchor, VCG, and RCPT share the same target, expert evaluations, integration grid, ridge convention, and random streams. A schedule fitted on 16 particles is frozen before 2,000 fresh Proposal-Only or SMC trajectories are generated in the primary comparison (Table 2). All runs use 200 integration steps, ridge10 −4 , SMC ESS threshold0.7, and the expert scheduledefault_linear / 1-t ** 2 / cos_t. The calibration ablation uses these same seeds and deployments. Its residual diagnostic is an active-path integral, not an independent target-law risk estimate. Illustrative replay and calibration ablation. The main Checker figure uses one seed, 16 fitting particles, 200 integration steps, and the exact common-endpoint scheduledefault_linear / 1-t ** 2 / cos_t. Each frozen schedule generates 10,000 particles. Endpoint estimates use fixed 2,000-particle audit indices; Proposal-Only has no FK weights, resampling, or terminal reweighting. The accompanying SMC replay retains correction. 27 Table I.1. Single-seed Checker figure audit. This replay is illustrative, not an additional ten-seed estimate. Residual is an active-path integral and is not pooled across deployment modes. ModeController W 2 MMDResidual ProposalAnchor0.76890.0645668.21 ProposalVCG1.19330.25723087.30 ProposalECG1.31560.4725291.44 ProposalRCPT0.68190.067030.03 SMCAnchor1.38880.138748.64 SMCVCG5.72076.94635673.08 SMCECG1.76220.4858152.01 SMCRCPT0.94450.123540.87 Table I.2. Checker calibration ablation. The same ten paired seeds, 16 fitting particles, fixed schedule, and 2,000-particle deployments as Table 2. Values are means. Direct LOO replaces the structured amplitude criterion. Residual is the integrated active-path diagnostic, not independent target-law risk; its Direct LOO Proposal-Only mean is driven by one severe run. Ancestors counts terminal SMC lineages out of 2,000; without resampling every trajectory retains its own ancestor. ModeController W 1 ↓ W 2 ↓ MMD↓ Residual Ancestors Proposal Anchor0.474 0.6460.048614.372000.0 Direct LOO 0.465 0.6360.0681777.692000.0 RCPT0.4420.5750.073103.242000.0 SMCAnchor0.367 0.5100.04831.56338.7 Direct LOO 0.435 0.6310.05925.62630.7 RCPT0.3160.4640.04225.09568.5 In this calibration ablation, RCPT has lower meanW 1 andW 2 than Direct LOO in both deployment modes and lower mean MMD under SMC. Direct LOO has lower Proposal-Only MMD and retains more SMC ancestors on average (630.7 versus 568.5). These distinct rankings separate endpoint accuracy from genealogy. I.2 Common-fit realized-ray and adaptive-regularization audit Protocol and evaluation law.This audit isolates the calibration decision on identical fitting states. Thirty raw-VCG pilot fits use the 12-field Checker dictionary, fitting populationsN ∈8, 16, 32, and ten paired seeds per budget. At each of 200 native steps, RCPT and Direct LOO use the same fitted direction. LOO-selected ridge and PCR use the same fitting states; PCR recomputes its basis within deletion folds. Selection uses fitting residuals only. No endpoint metric selects a ridge level, rank, or amplitude. Coefficient clipping and calibration fallback are disabled in this audit, so scalar rules remain on the fitted ray. This is a separate cohort from the primary fixed-schedule comparison and its matched calibration ablation. Independent reference states separate calibration from evaluation. Intermediate normalized target densities are un- available for the learned expert paths. We therefore use two independent corrected Anchor-SMC systems of 16,384 particles each, with separately normalized weights. Both systems resample five times; minimum pre-resampling ESS is 9,102–9,229. Their combined 32,768-particle weighted lawμ t defines the reference risk. This is neither exact target integration nor iid target sampling. Disagreement between systems measures simulation sensitivity; it does not bound shared model bias or the target-reference covariance gap in Appendix H.4. The audit uses 21 fixed rounded indices fromlinspace(0, 199, 21). Each native step is assigned to its nearest audit node; the resulting cell counts divided by 200 give time weightsw k . Reported Risk/Anchor is the ratio of P k w k Var μ t k (g 0 t k + a ⊤ t k λ t k ) to the corresponding Anchor risk, summarized over the ten fitting seeds. Bootstrap intervals resample paired fitting seeds, not time nodes. They describe variation over fits conditional on this reference construction. Table 1 in the main text reports this comparison. 28 Adaptive regularization. RCPT has the lowest mean reference risk among the implemented rules at all three fitting populations (Table 1). AtN = 16, its paired Risk/Anchor differences from Direct LOO, ridge, and PCR are −0.6331,−0.3375, and−0.6804, with respective 95% bootstrap intervals[−1.363,−0.0669],[−0.7484,−0.0510], and[−1.182,−0.2518]. AtN = 8and 32, the intervals against Direct LOO and ridge include zero; the intervals against PCR remain below zero. These are pairwise intervals over ten fits, not simultaneous guarantees. The results support scalar calibration at low particle budgets without establishing dominance over all directional estimators. PCR changes a learned basis and rank; it does not exhaust grouped or coordinatewise shrinkage. Realized calibration. For each fit and node, replacepbyμ t in Eq. (D.1) to obtainq μ ,κ μ , and the reference-ray oracle. Direct variance evaluation agrees with the quadratic atα ∈ 0, 0.25, 0.5, 0.75, 1to maximum absolute error2.328× 10 −10 . None of the 630 fit–time cells requires the near-zero-curvature convention, whose tolerance is max10 −14 , 10 −10 max(V μ (0), 1). The structured ingredients are not quantitatively calibrated pointwise. MedianS/q μ rises from0.105to0.276and0.606 asNgrows;κ μ < 0in 56/630 cells (Table I.3). A nonnegative penalty cannot match those negative covariances exactly. Nevertheless, mean integrated ray regret is8.359,4.500, and3.776for RCPT, versus12.900,27.960, and4.635for Direct LOO. Direct LOO has smaller mean absolute amplitude error atN = 8and 32, but not smaller regret. Curvature weights amplitude errors differently across times, and boundary optima add a one-sided term. This distinction explains why unweighted amplitude accuracy and decision quality need not have the same ordering. Table I.3. Realized quadratic diagnostics. All quantities use the independent corrected-SMC reference law. Spearman coefficients summarize the seed-levelS–q μ and b P–κ μ diagnostics. Their limited agreement precludes interpreting the structured ingredients as accurate pointwise estimates. NMedian S/q μ κ μ < 0 ρ(S,q μ ) ρ( b P,κ μ ) 80.1054.8%0.4180.406 160.2766.2%-0.1880.103 320.60615.7%0.1150.673 What the oracle calculation establishes.The expected-risk oracle and the realized calibration bound answer different questions. Applying theH,P,Bidentity to the empirical distribution of ten fits gives an oracle gain over the better endpoint larger than the two-system sensitivity in 22/63 population-size–time cells: 6/21, 6/21, and 10/21. This diagnoses Theorem 2’s expected-risk calculation on the reference law. It is not the frequency with which Theorem 3’s plug-in error bound is informative. The latter additionally requires control of realized approximation errors and target-reference transfer. No finite-sample certificate for the implemented selector follows from this audit. Endpoint deployment and completeness.Each frozen schedule generates 10,000 paired trajectories in Proposal-Only and corrected-SMC modes. Exact discrete optimal transport estimatesW 2 on a predeclared paired 2,000-sample metric subset. Table I.4 retains all attempted denominators. RCPT completes 60/60 deployments, compared with 45/60 for VCG and 59/60 for PCR. Means for incomplete arms condition on completion and are not unconditional performance comparisons. Residual improvement does not uniformly improve terminal transport. RCPT has lower mean Proposal-OnlyW 2 than Direct LOO, ridge, and PCR at each budget, but SMC favors ridge atN = 16and both ridge and PCR atN = 32. Paired RCPT-minus-AnchorW 2 intervals include zero in Proposal-Only at every budget and in SMC atN = 8, 16. At N = 32, the SMC difference is+0.0817, with 95% bootstrap interval[+0.0051, +0.1629]. Reference risk, propagated transport energy, and corrected finite-particle endpoints are different quantities; the one-sided transport bounds do not order these observed metrics. Selection cost.Mean selector time per 200-step schedule is0.132seconds for RCPT,1.31for Direct LOO,9.34for ridge selection, and13.03for PCR. These are selector timings within this implementation, not total sampling times or hardware-independent complexity comparisons. Together with the risk results, they support scalar shrinkage as an inexpensive decision under limited fitting information. They do not establish its global optimality among regularizers. 29 Table I.4. Common-fit Checker endpoints. MeanW 2 ±standard error across completed seeds. Finite gives the completion count over every planned seed; lineages are SMC terminal ancestors out of 10,000, averaged over completed runs. The reference-risk analysis includes all 30 fits per method, including those whose endpoint deployment failed. Proposal-OnlyCorrected SMC N ControllerW 2 ↓ FiniteW 2 ↓ Finite Lineages 8Anchor 0.7725± 0.0136 10/10 0.5773± 0.0290 10/101723 VCG 1.594± 0.17795/10 2.316± 0.31448/10613.9 Direct LOO 0.9814± 0.0422 10/10 0.7267± 0.0735 10/101642 LOO ridge 1.082± 0.0467 10/10 0.7458± 0.0871 10/101631 LOO PCR 1.073± 0.07559/10 0.8080± 0.1280 10/101559 RCPT0.8360± 0.044710/100.6417± 0.037210/101873 16 Anchor 0.7973± 0.0159 10/10 0.6602± 0.0337 10/101893 VCG 1.423± 0.59093/10 3.744± 0.25029/10657.2 Direct LOO 0.9174± 0.1168 10/10 0.7979± 0.1391 10/101542 LOO ridge 0.9675± 0.1051 10/10 0.6963± 0.0639 10/101832 LOO PCR 0.9682± 0.1157 10/10 0.7585± 0.0634 10/101804 RCPT0.9062± 0.109710/100.7341± 0.101110/101592 32 Anchor 0.8044± 0.0188 10/10 0.5990± 0.0310 10/101895 VCG 0.7991± 0.0545 10/10 1.731± 0.5101 10/102400 Direct LOO 0.7918± 0.0486 10/10 0.6553± 0.0218 10/102861 LOO ridge 0.8054± 0.0455 10/10 0.6068± 0.0463 10/103203 LOO PCR 0.8023± 0.0390 10/10 0.6067± 0.0463 10/103138 RCPT0.7851± 0.048210/100.6806± 0.029910/102671 I.3 Scaffold protocol The fixed benchmark contains all 76 CrossDocked2020 scaffold–pocket tasks [16]. Its four pretrained expert paths are EDM for the fragment and full ligand, GeoDiff for fragment conformations, and pocket-conditioned DiffSBDD for the full ligand [20, 34, 48]. For each expert velocity v i , the controller contains φ i,0 = v i , φ i,1 = z i v i , φ i,2 = 1 2 (3z 2 i − 1)v i , r i = logq i − I −1 I X ℓ=1 logq ℓ , z i = tanh r i − μ i σ i , (I.2) giving 12 relative-density-gated fields with bounded gates. The degree-zero fields are exactly the original mechanistic expert velocities. The gate statistics are fixed from the fitting cohort and never selected using decoded chemistry. Each pocket uses 16 fitting particles with one fitting seed. Anchor is the natural ACE proposal; VCG is the ordinary 12-field regression; RCPT applies exact weighted leave-one-out refitting at the same fitted times. The resulting schedules are frozen and deployed on 16 fresh paired trajectories per arm. Proposal-Only deployment retains neither correction weights nor resampling. RDKit decoding, scaffold retention, uniqueness, diversity, QED, SA, Lipinski summaries, and QuickVina docking are computed only after deployment and never enter controller selection. Affinity-improved yield uses the fixed generation denominator; “unique useful” additionally deduplicates valid, scaffold-retaining, reference- beating molecules. Docking success alone counts any candidate with a successful docking result, even if it fails the separate chemical-validity check, divided by all generated slots. Paired pocket bootstrap intervals treat the 76 pockets as statistical units. The main table uses the original complete fit-16 deployment; an independent 64-trajectory replication reuses the same frozen schedules and increases evaluation to 4,864 candidates per arm. The common-state audit evaluates all 76 fitted schedules on identical shared reference states, rather than their active deployment paths. VCG exceeds Anchor risk in 75/76 pockets. RCPT improves on VCG in all 76 and repairs 72 of those 75 harmful updates; Direct LOO repairs 73. Mean Risk/Anchor is 25.9641 for VCG, 0.8935 for RCPT, and 0.7958 for Direct LOO. Direct LOO beats RCPT in 73/76 pockets; the paired RCPT-minus-Direct difference is 0.0977 (95% bootstrap CI [0.0622,0.1197]). The worst ratios are 1491.65, 2.00, and 3.02, respectively: the mean improvement and tail comparison differ. 30 Table I.5. High-precision 76-pocket scaffold deployment. The fixed fit-16 schedules are independently deployed with 64 candidate slots per arm and pocket (4,864 per arm). This table isolates the VCG-to-RCPT comparison; downstream means remain close to Anchor. Risk is normalized by Anchor. Dock counts successful docking over all generated candidates, irrespective of chemical validity. Bold marks the better of the two reported controllers. Controller Validity↑ Scaffold↑ Risk/Anchor↓ Dock success↑ Affinity↑ Unique useful↑ VCG0.2160.21223.8220.5690.1480.101 RCPT0.4330.4150.8780.9040.2920.224 Table B.3 reports this comparison. Direct LOO lowers active-path Risk/Anchor by 0.1199 relative to RCPT (95% CI [−0.1293,−0.1099]), but its validity difference is−0.0206([−0.0485, 0.0074]) and unique useful yield difference is−0.0148([−0.0419, 0.0115]). All reported downstream paired intervals include zero. The molecular implementation includes coefficient clipping and an empirical fallback, so these deployed arms include the guards in Appendix H. The bundled three-later-seed Direct summary reports validity 0.4252, scaffold yield 0.4068, Risk/Anchor 0.7576, docking 0.9191, affinity yield 0.2944, and unique useful yield 0.2536. It is a different aggregation from Table B.3; matched three-seed control summaries are unavailable, so it is not used for paired comparisons. The separate ECG run covers the same 76-pocket manifest with 16 fitting particles and 16 fresh outputs per pocket. The implementation requires thescoresbasis:φ i = logq (i) t and∇φ i = ∇ logq (i) t , one potential per expert. It solves the weighted Ritz system with centered potential–response covariance and the uncentered gradient Gram matrix, then applies the configured coefficient clip. It does not reject a finite ECG coefficient merely because its empirical VCG residual increases. This four-potential class differs from the 12 gated velocity fields; the endpoint comparison therefore assesses the instantiated proposal methods, not two optimizers on an identical field class. Table B.2 reports the separate controller-capacity ablation. Against VCG-4, RCPT improves validity by0.0510at fit 16 (paired 95% bootstrap interval[0.0041, 0.0987]) and0.0872at fit 64 ([0.0395, 0.1349]). At fit 64, its affinity-qualified and unique useful yields improve by0.0699([0.0329, 0.1086]) and0.0650([0.0288, 0.1020]), respectively. These gains establish downstream benefits beyond reducing residual risk; VCG-4 actually has lower mean residual risk. At fit 16, the unique useful yield gain is 0.0255 with interval [−0.0148, 0.0666], so that particular gain is not statistically resolved. I.4 QM9 steering protocol The pretrained EEGSDE model is conditioned on dipole moment and steered toward polarizability and HOMO–LUMO- gap targets. The 12-field class crosses the two reward-gradient families with bounded time/state Legendre gates; all arms use the same model, target, evaluator, resampling rule, and total trajectory budget. Reported properties come from independently trained evaluators, not the guidance networks. Molecular identities are RDKit canonical isomeric SMILES. The split protocol uses 16 particles to fit one schedule and eight independent SMC deployment batches of 16, for 144 trajectories in total. The online protocol uses one live population of 144: coefficients are refitted from that current population, and the same particles continue and become outputs. Fitting uses all 144 current particles before the next stochastic transition. The serializedfit_particles=16option is inactive in online mode. Hence online is a strong adaptive batch-generation baseline, whereas split measures out-of-fit reuse of a frozen controller. Minimum ESS and surviving lineage counts are divided by batch size in the combined table, but one 144-particle genealogy is not statistically identical to eight independent 16-particle genealogies. Both protocols contain eight paired fit seeds and use leave-one-out refitting (mean trust 0.203 split and 0.945 online). ECG proposal comparison. The frozen ECG baseline uses the scalar potentials φ r,ℓ (x,t) = ν t Z e r (x,t) 0 −sP ℓ (tanhs) ds, r ∈polarizability, gap, ℓ = 0,..., 5,(I.3) wheree r is target minus prediction in normalized property units andν t is the one-step reverse-transition variance. Thirty- two-node Gauss–Legendre quadrature evaluates the potentials; analytic gradients supply the 12 gated fields. Potentials 31 Table I.6. QM9 steering split means±sample SD over eight paired fits. Unique success counts distinct valid identities meeting both targets at the unchanged one-MAD threshold per 128 generation slots. Minimum ESS and lineages are reported per 16-particle deployment batch. Split methodEval. MAE Unique success /128Min ESSLineages Anchor0.9171± 0.041320.75± 6.27 4.023± 0.656 5.297± 1.44 VCG0.9281± 0.11413.88± 5.792.624± 1.14 2.578± 1.49 RCPT0.8997± 0.074623.00± 8.054.207± 1.545.641± 2.65 and fields share fit-batch RMS normalization. At each fit,( b G + 10 −8 I)θ = d Cov w (g 0 ,φ), with b G ij = E w [∇φ ⊤ i ∇φ j ]. Coefficients are clipped to±2× 10 −4 without a residual-risk acceptance rule. Sixteen particles fit a frozen schedule before eight independent 16-particle deployment batches, using eight seeds and 1,000 steps. Table I.7. ECG completion and diversity diagnostics. Frozen/deploy values use eight seeds;±denotes sample SD. The reduced online DriftLite-style configuration has one completed seed. ESS and lineages are absolute counts per batch. Success is distinct from identity-level unique success. ECG configurationSeedsMAE↓Success↑Valid Unique Min ESS Lineages Frozen/deploy, 12 fields8 0.8500± 0.1858 0.4049± 0.2276 0.9150 0.07611.01.0 Reduced online11.24140.0000 1.0000 0.03471.01 ECG’s property scores conceal severe collapse in frozen deployment. Every batch retains exactly one ancestor. Mean uniqueness is 0.0761, and mean resampling count is 1,105.5 per seed across eight batches, or 138.19 per batch. The reported deployed residual ratio is approximately3.58× 10 5 . The 0.4049 evaluator success statistic counts repeated outputs; it cannot be substituted for the number of distinct successful molecules. Table 4 therefore reports distinct successful identities and lineage fractions, while Table I.6 retains the available identity-level counts for Anchor, VCG, and RCPT. The original 12-field online ECG did not complete because its controller geometry became nonfinite. A reduced implementation closest to the DriftLite online construction completed one seed, shown in Table I.8. Its error exceeds Anchor and it has zero success. This is a configuration-specific feasibility result, not an eight-seed estimate or a general ordering of ECG and VCG objectives. Table I.8. Completed online ECG audit. One paired seed,N = 144. The Anchor, VCG, and RCPT values come from the same evaluation run; the reduced ECG result is from the accompanying online audit. The failed 12-field ECG run is not assigned a finite score. ControllerEval. MAE↓Success↑Uniqueness↑ Anchor1.13070.00000.0069 VCG0.84220.37361.0000 ECG1.24140.00000.0347 RCPT0.83790.37881.0000 Online calibration ablation.Direct LOO replaces the structured amplitude in the same 12-field online pipeline. Its eight-seed cohort has no matched frozen/deploy run. Table B.4 isolates this change from the main proposal comparisons. One Direct LOO seed resamples once and retains 84 of 144 ancestors; the other seven retain all 144. Direct-minus-RCPT MAE is 0.0025 (95% paired bootstrap CI [−0.0017, 0.0074]). The endpoint difference is unresolved at this precision. Runtime is not compared across separately executed cohorts. Table B.4 reports the online calibration ablation. 32 Regularized Direct ablation.Write the exact deleted-loss quadratic asL D (α) = A + 2bα + cα 2 and the structured amplitude as α S . The exploratory selector minimizes α R = arg min α∈[0,1] L D (α) + γc(α− α S ) 2 .(I.4) Forc > 0, its unconstrained solution is(−b/c + γα S )/(1 + γ); clipping to[0, 1]is performed once. The run uses γ = 1 and disables componentwise coefficient clipping. Numerically vanishing curvature selects Anchor. Table I.9. Exploratory regularized Direct LOO. One fresh online seed,N = 144. This tests an additional regularized selector, not the main RCPT method; its small error advantage is not replicated evidence. ControllerMAE Success Unique /144 Resamples Ancestors Anchor1.32440.00000701 VCG0.81320.4541450144 Direct LOO0.80800.4441460144 Reg. Direct (γ = 1) 0.80440.4538460144 RCPT0.80990.4594460144 I.5 CIFAR-10 class-conditional image generation Mechanism and physical deployment. The pretrained DLSM models are fixed. The unit-power mechanism study usesβ = γ = 1, all ten classes, and five fitting seeds. Nested fitting cohorts contain 8, 16, or 32 forward-noised training images at 16 midpoint times; independent test cohorts contain 128 states per class and time. Models, states, and realized Hutchinson probes are paired across methods. The corrected response is Eq. (H.4). This audit is separate from the β = γ = 2.5 terminal cohort. Table I.10. CIFAR-10 unit-power mechanism study. Fifty class–seed cells per fit budget,β = γ = 1. Risk is held-out Risk/Anchor; harm means a ratio above one. RCPT improves on VCG in 46/50, 47/50, and 43/50 cells at fits 8, 16, and 32. FitControllerMean Risk/Anchor↓Harm↓ 8VCG1.14046/50 8RCPT0.99713/50 16VCG2.16446/50 16RCPT1.09022/50 32VCG2.69536/50 32RCPT1.52223/50 RCPT reduces mean risk and the number of harmful cells at all three budgets. Protection remains incomplete: its mean risk exceeds Anchor at fits 16 and 32. A separate five-seed, fit-16 audit matches the terminal target and Anchor (β = γ = 2.5; Table B.6). RCPT has only 5/50 harmful cells, compared with 22/50 for VCG-9 and 13/50 for Direct LOO, and the smallest worst ratio among fitted methods. VCG-4 has the lowest mean risk; RCPT does not improve on VCG-9’s mean in this setting. All deviations from Anchor are small. These common-state surrogate risks describe behavior under the terminal configuration; they are not pooled with the unit-power audit. Metrics and cohort boundaries.Each generation cohort contains 32 images per class and seed. A separate ResNet-20 supplies independent accuracy and classwise polynomial-kernel KID against 256 test images per class. DLSM accuracy uses the guidance classifier itself, so it measures agreement with the guider rather than independent class fidelity. We therefore use ResNet accuracy as the independent classification test and interpret DLSM accuracy as a guider self-consistency diagnostic. Clean-FID uses Inception-v3 and all 10,000 test images. DINOv2-small uses normalized CLS features: FD is pooled, and classwise RBF MMD uses a target-only median-distance bandwidth. CLIP ViT-B/32 class margin is target similarity minus the largest non-target similarity, averaged over four fixed prompt templates. None enters controller fitting. Pooled FD/FID and means of per-seed FD/FID differ substantially at these sample sizes and are kept apart. 33 Image-generation protocol.The terminal comparison uses seed 261012, target powerβ = 2.5, and matched Anchor guidanceγ = 2.5. Each method generates 32 outputs per class, or 320 images. The protocol uses fit 16/evaluate 128 states at 16 schedule nodes and PFO-500, with no Langevin corrector. All methods retain the same fixed+1target-class coefficient on the scale-1.5 base; VCG-9, Direct LOO, and RCPT fit the remaining nine auxiliary fields. Thus the Anchor–RCPT comparison isolates the fitted auxiliary correction under a shared target-class boost. VCG-4 restricts the fitted correction to four fitting-covariance directions. Table B.5 reports the calibration ablation; Table B.6 provides the replicated capacity diagnostic. The single-seed endpoint comparison does not estimate between-fit uncertainty. Endpoint interpretation. RCPT improves six of seven metrics over matched Anchor and ties DLSM accuracy. Independent accuracy increases from 0.6625 to 0.6750 (four additional correct predictions), KID falls from 0.2688 to 0.2407, and Clean-FID falls from 89.043 to 88.232. RCPT also improves six metrics over VCG-9, which has higher DLSM accuracy. Relative to Direct LOO, RCPT has higher independent accuracy and CLIP margin and lower Clean-FID; Direct LOO has better DLSM accuracy, KID, DINO FD, and classwise DINO MMD. Thus RCPT improves a broad range of endpoints without uniformly dominating the direct calibration ablation. These are observed single-seed comparisons, not seed-level significance claims. FID and DINO FD compare Gaussian feature summaries; they are not the image-space W 2 in Theorem 4. Qualitative contrasts. Figure I.1 shows three class–sample pairs from the terminal cohort. RCPT’s evaluator prediction matches the target in each example, while Anchor and VCG-9 do not. The panel illustrates the corresponding classification changes. pred: deer Target: frog #29 Anchor pred: deer VCG-9 pred: deer Direct LOO pred: deer VCG-4 pred: frog RCPT pred: dog Target: cat #29 pred: dogpred: catpred: deerpred: cat pred: bird Target: airplane #23 pred: birdpred: airplanepred: dogpred: airplane Figure I.1. CIFAR-10 classification contrasts. Three class–sample pairs from seed 261012 under the matchedβ = γ = 2.5 protocol. Labels are independent evaluator predictions. Direct LOO is the calibration ablation; VCG-4 is the capacity ablation. Aggregate endpoint metrics use all 320 outputs. I.6 Ten-pocket scaffold fixed-rule comparison We compare eleven prespecified controllers on the same ten scaffold–pocket targets. Every arm uses oneN fit = 16 fit (one fitting seed) and a paired, independentN deploy = 16Proposal-Only deployment per pocket. Anchor, VCG, and RCPT are the immutable matched rows from the full-76-pocket study; the eight fixed-scalar, ridge, EMA, and norm-cap controls reuse the same fitted information and deployment seeds. No decoded molecule or docking result tunes a controller. The ten pockets are the equally weighted experimental units. 34 Table I.11. Scaffold simple-baseline comparison. Equal-pocket means over the matched ten-pocket fit-16/deploy-16 subset. “Affinity” is the fixed-denominator yield that both retains the scaffold and beats the pocket reference; “unique useful” additionally deduplicates molecular identities. Risk is measured on each method’s active deployment path because the optional common-state audit did not complete; it is therefore a mechanism diagnostic, not a same-state counterfactual. Bold marks the best value within each of the two controller groups, including ties. ControllerRisk/Anchor Valid Scaffold Affinity Unique useful QEDSA Best QVina ↓ ↑ ↑ ↑ ↑ ↑↓ Anchor1.000 0.4630.4500.3630.263 0.517 0.653−7.95 VCG13.895 0.3190.3190.2940.200 0.409 0.584 −7.66 RCPT0.8900.5630.5130.4130.3060.4890.662−7.93 Fixed α = 0.251.915 0.5630.5440.4560.356 0.548 0.663 −7.94 Fixed α = 0.505.116 0.5880.5690.5060.406 0.529 0.629−8.15 Fixed α = 0.7511.483 0.4060.4000.3500.256 0.509 0.631 −8.10 Ridge ρ = 10 −3 18.750 0.2810.2810.2560.163 0.475 0.593 −7.87 Ridge ρ = 10 −2 18.429 0.2380.2380.2310.138 0.483 0.593 −7.75 Ridge ρ = 10 −1 20.740 0.2750.2750.2380.144 0.382 0.548 −7.69 VCG+EMA β = 0.92.991 0.2690.2690.2440.150 0.441 0.596 −7.53 VCG+norm cap 0.250.994 0.5440.5380.4560.356 0.507 0.664 −8.02 Table I.11 confirms that RCPT repairs VCG: validity rises from 0.319 to 0.563, scaffold yield from 0.319 to 0.513, and active-path risk falls from 13.895 to 0.890. It is not uniformly best among simple rules on this small subset. Fixed α = 0.50has higher scaffold, affinity-improved, and unique-useful yields, with paired bootstrap intervals excluding zero, while the norm cap is numerically strong on several endpoints. Conversely,α = 0.75, every ridge setting, and EMA deteriorate substantially. The result supports adaptive negative-transfer protection, not dominance over an oracle-chosen fixed scalar for every subset. I.7 QM9 steering fixed-rule comparison The reward audit uses four seeds, 1,000 integration steps, 16 fitting particles, and eight 16-particle corrected deployment microbatches (128 slots per controller and seed). It uses the normalized three-field class consisting of the two reward- gradient fields and reverse base drift. The first deployment microbatch shares the fitting random stream and the remaining seven are fresh, so this audit supplements rather than replaces the independently seeded eight-seed, 12-field primary result in Table 4. Table I.12. QM9 steering simple-baseline comparison. Means over four fit seeds; MAE reports mean±seed standard deviation. Unique success counts distinct valid molecular identities meeting both targets at the unchanged one-MAD threshold per 128 generation slots, so duplicates and invalid samples consume budget. Risk is the integrated selected/Anchor residual-variance ratio on each controller’s own deployment genealogy and is not a common-state comparison. Bold marks the best value within each of the two controller groups, including ties. ControllerRisk/Anchor↓MAE↓Unique success /128↑ Anchor1.0000.9346± 0.033120.5 VCG0.2420.8499± 0.020533.5 RCPT0.2240.8252± 0.046239.2 Fixed α = 0.251.2150.9678± 0.099620.5 Fixed α = 0.500.3300.8781± 0.029829.2 Fixed α = 0.750.0860.8593± 0.059735.2 Ridge ρ = 10 −3 0.1570.8892± 0.034131.8 Ridge ρ = 10 −2 0.0680.8237± 0.043838.5 Ridge ρ = 10 −1 0.041 0.8136± 0.063839.8 VCG+EMA β = 0.91.7460.8637± 0.063932.2 VCG+norm cap 0.250.0450.8412± 0.041333.2 Table I.12 shows a complementary boundary. No fixed scalar improves mean MAE over VCG, whereas RCPT lowers 35 MAE from 0.8499 to 0.8252 and raises unique success from 33.5 to 39.2 while slightly lowering active-path risk. Fixed ridge atρ = 10 −1 leads MAE and unique success, but its effect is nonmonotone:ρ = 10 −3 is worse than VCG in both metrics. With only four seeds, all paired MAE intervals against VCG include zero; this audit therefore establishes a fixed-rule sensitivity, not statistically decisive reward-domain superiority. I.8 Bias and penalty-estimator audits Coefficient bias. The expected-risk oracle retainsB = (λ ⋆ ) ⊤ C E[ b λ− λ ⋆ ]; the structured model does not estimate it separately. The Checker audit uses the 12-field basis,N = 8, 8,192 reference particles, 10,000 independent fitting ensembles, 80 time points, and the matched ridge convention. The mean bias-aware and no-bias trust are 0.1900 and 0.1934; their mean absolute pointwise difference is 0.0176 (median 0.0077, 95th percentile 0.0661, maximum 0.1071). IgnoringBincreases the integrated oracle risk by 0.781%. This supports a practical simplification on Checker, not the general claim B = 0 or B = o(N −1 ). Penalty approximation. The large-population Checker audit uses five trajectories, 8,192 population particles per trajectory, 40 fitted times, and 500 repeated fits atN ∈ 16, 32, 64, 128, 256. Across 200 seed–time problems, the structured proxy andN −1 tr(ΩC † )have Spearman correlation 0.893, but their integrated scale ratio is only 0.167. AtN = 256, measured penalty is1.071×theΩ-based prediction while the structured proxy is low by6.289×. At N = 16, neither first-order approximation is quantitatively accurate. Realized calibration is a separate empirical question. Neither preceding expected-penalty audit estimates the realizedκrequired by Theorem 3. The corresponding target-law quantities areq = Var p (a ⊤ b λ) ,κ = Cov p (g 0 + a ⊤ b λ,a ⊤ b λ) , andα ray = Π [0,1] (1 − κ/q)whenq > 0. The relevant errors areS − qand b P − κ, together with selected-amplitude error, realized ray regret, and negative-transfer frequency. Whenκ < 0, the ray oracle retains the full update, and a nonnegative penalty has error at least−κ. Appendix I.2 measures these quantities on an independent corrected-SMC reference. It finds lower selected-ray regret despite substantial proxy error. The reference is approximate and dependent within each particle system, so the audit does not certify the target-law bound for Algorithm 1. Table I.13. Covariance-penalty ablation. Ten paired Checker seeds. Replacing the structured effective-rank proxy with a fullΩ estimate gives no consistent endpoint gain. This changes the penalty estimator, separately from the Direct LOO amplitude ablation. ModeFitEstimator W 2 ↓MMD↓ SMC8Full Ω2.0102.917 SMC8RCPT1.7843.181 SMC16Full Ω2.1702.213 SMC16RCPT2.0302.069 Proposal8Full Ω1.7692.519 Proposal8RCPT1.7682.519 Proposal16Full Ω1.4691.275 Proposal16RCPT1.4431.269 J Additional Mechanism Diagnostics The diagnostics below supplement the main figures with fitting-budget, paired-risk, and genealogy measurements. J.1 Mechanism diagnostics The fitting-population sweep contains ten paired Checker SMC runs at each ofN fit ∈4, 8, 16, 32, 64, 128, 256, with 10,000 evaluation particles from a fresh deployment population. All 70 runs per method completed. Figure 1 compares the integrated fitting and independent residual ratios, each with its own matched Anchor denominator. Low-particle VCG nearly interpolates its fitting states, although its independent risk can be orders of magnitude larger. RCPT gives up some fitting improvement to reduce independent risk. At fit 8 the full sweep has median risks 105.22 and 0.966, and 36 mean risks 524.17 and 1.084 for VCG and RCPT; the mean reflects a remaining RCPT tail. This fitting-population sweep is separate from the fixed-schedule endpoint comparison in Table 2. Figure J.1 displays the same sweep against particle count on both panels. The median support ratiosN fit /r eff for RCPT are 1.33, 1.15, 1.48, 2.80, 5.50, 10.88, and 21.75; the effective rank itself changes at small N . Mean amplitude rises from 0.143 atN = 4to 0.921 atN = 256, while VCG and RCPT independent risks converge. The method therefore retains more of the fitted update as the independent risks converge; this observation does not identify the population-optimal headroom. 48163264128256 Fitting particles N fit 10 0 10 1 10 2 Independent risk / Anchor (a) Out-of-fit risk Anchor VCG RCPT 48163264128256 Fitting particles N fit 0.0 0.2 0.4 0.6 0.8 1.0 Mean amplitude α (b) Selected trust Figure J.1. Increasing fitting information closes the gap to VCG. Risk curves show medians and interquartile ranges; amplitude shows mean and standard error over ten seeds at each particle count. Figure J.2 compares independent risk across the paired methods. RCPT improves 66/70 Checker runs and repairs 34/47 harmful VCG runs to Anchor or below. The separate Scaffold common-state audit contains 75 pockets; pocket 45 was reserved for the engineering smoke test before this audit. RCPT improves every pocket and repairs 68/73 harmful VCG schedules. This diagnostic cohort is distinct from the full 76-pocket deployment table. The ratios are not pooled across domains. 10 −1 10 0 10 1 10 2 10 3 10 4 VCG risk / Anchor 10 −1 10 0 10 1 10 2 10 3 10 4 RCPT risk / Anchor Harmful: 47/70 Repaired: 34/47 Checker SMC (n= 70) 10 −1 10 0 10 1 10 2 10 3 VCG risk / Anchor 10 −1 10 0 10 1 10 2 10 3 RCPT risk / Anchor Harmful: 73/75 Repaired: 68/73 Scaffold (n= 75) 0.0 0.2 0.4 0.6 0.8 1.0 Mean RCPT amplitude α Figure J.2. Risk repair and retained adaptation. Points below the diagonal favor RCPT. Points to the right of the vertical Anchor line and below the horizontal one are harmful VCG fits repaired to Anchor or below. Color denotes mean selected amplitude, not terminal quality. J.2 Selected amplitude across tasks Figure 4 summarizes time-resolved amplitudes. Thick lines are means across units, bands are interquartile ranges, and thin lines are individual schedules. The summary averages the saved time nodes without claiming a common continuous-time integral. Native time runs in opposite generation directions in some domains. Amplitude is relative to each fitted coefficient; it is not a comparable drift norm or an independent measurement of population headroom. Figure 4 contains 76 Scaffold pockets, eight online QM9 seeds with 144 live particles, ten Checker schedules, and ten CIFAR schedules. The updated Checker panel shows strong early retention followed by smaller amplitudes, while the CIFAR panel is mostly conservative. The CIFAR panel displays ten schedules; the replicated risk audit aggregates 50 class–seed cells. These curves describe the rule’s response within each plotted cohort; they do not identify a terminal-optimal amplitude. 37 0.00.20.40.60.81.0 Generation progress (noise to data) 0.0 0.2 0.4 0.6 0.8 1.0 Weight ESS / N (a) Current weights Anchor VCG Structured RCPT 0.00.20.40.60.81.0 Generation progress (noise to data) 1/144 0.1 1 Surviving lineages / N (b) Genealogy Figure J.3. Weight ESS and genealogy in online QM9 steering. Median and interquartile range across eightN = 144seeds, ordered in generation time. Anchor resamples 85.8 times on average and ends with one ancestor in every seed; VCG and RCPT never resample and retain all 144. Resampling can restore weight ESS without restoring lost ancestry. These are observed path-dependent outcomes, not a universal consequence of local residual ordering. J.3 Residual risk as a terminal surrogate Across 83 finite Proposal-Only profiles, integrated held-out residual has Spearman correlation0.775with empirical TV (Pearson0.881after log scaling its many-order-of-magnitude range), while the theorem-alignedB 1 = 1 2 R 1 0 √ V t dthas Spearman correlation0.730. Retaining the time-resolved profile matters:B 1 is a median47%tighter than the additional time–Cauchy relaxationB 2 = 1 2 q R 1 0 V t dt . Nevertheless, every evaluatedB 1 exceeds one and its median is5.99× empirical TV. We therefore use residual risk as a correlated comparative diagnostic, not an exact terminal error estimate. J.4 Cross-domain portability and fixed controls The CIFAR-10 study in Appendix I.5 extends the finite-particle mechanism to a pretrained image model. Its independent- state risk audits and separate terminal cohorts answer different questions: residual generalization and downstream behavior. The current terminal comparison uses one seed with matched target power and Anchor guidance 2.5. Appendix Tables I.11 and I.12 compare fixed shrinkage, ridge, and engineering safeguards. The best fixed heuristic does not transfer: a fixed scalar can be strong on selected scaffold yields, while no fixed scalar improves QM9 steering MAE over VCG; ridge behavior is similarly task-dependent and nonmonotone. RCPT is the common arm that estimates trust from the available fit rather than selecting a task-specific regularization value. K Computational Cost After expert fields and divergences are available, VCG costsO(Nm 2 + m 3 )per fitted step. Leave-one-out refitting reuses those features and adds small dense linear algebra, but no expert, denoiser, or evaluator call. Table B.7 reports matched costs with hardware; all times exclude downstream docking and offline evaluation. Scaffold timing means cover all 76 pockets on an RTX A6000. QM9 times use four A100-SXM4 runs; runs on other hardware are excluded. The 1,000-step QM9 runs predated memory instrumentation, so peak memory comes from a matched 10-step probe. Small VCG–RCPT timing reversals reflect run variability. RCPT adds 3.13 seconds over VCG in approximately 3,880 seconds (0.081%) with equal probed memory; the recorded Anchor time is 3877.460 seconds. Why the small-particle regime matters computationally. RCPT’s main systems benefit is not that leave-one-out refitting makes one trajectory cheaper; it is that a controller can be fit on the small ensemble the user actually wants to generate. Table K.1 measures this scaling in our implementations. From 16 to 64 scaffold particles, allocated VRAM rises by3.93×and wall time by2.12×, whereas aggregate controller- solve time changes by only 0.20 seconds. On the same A100, matched QM9 steering probes require 4.569 GB at N = 16and 40.204 GB atN = 144(8.80×memory for9×as many live particles). These are direct measurements of particle scaling, not an extrapolation to thousands of particles. These matched measurements support calibration at low particle budgets: 12-field controllers fitted from 8–16 particles add negligible computation over VCG, while larger fitting populations increase model cost and memory. They do not 38 Table K.1. Particle scaling for the 12-field scaffold RCPT fit on NVIDIA RTX A6000 48GB GPUs, averaged over the complete 76-pocket benchmark. The Solve column is aggregate proposal-controller solve time; the remainder is dominated by trajectory and field/model computation. N fit s/pocketPeak GBSolve s 16197.4454.7405.282 32268.8529.3655.473 64419.19318.6115.478 measure a speedup over DriftLite’s published high-particle experiments [33]. A frozen schedule also amortizes its fitting time over independent outputs. Deployment remains microbatchable: it retains the fields needed to apply the combination while removing online covariance, LOO, and particle-synchronization work. 39