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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

arXiv:2610.04171v1stat.MLcs.AI

TL;DR

Finite-particle VCG can nearly eliminate fitting residual while increasing independent risk, making population guarantees insufficient to assess a fitted update. RCPT calibrates the retained update using deletion leave-one-out residuals; across molecular and image domains it mitigates harmful fits and improves terminal metrics relative to baselines including VCG.

  • Problem

    Finite-particle proposal fitting can fail to generalize: VCG may nearly eliminate fitting residual while increasing independent residual risk by orders of magnitude.

    Source: §1 · pp. 1–2

  • Method

    RCPT uses deletion leave-one-out residuals to estimate risk and calibrate the retained fraction of the VCG update, without additional model calls.

    Source: §1 · p. 2

  • Results

    Across molecular and image domains, RCPT mitigates harmful fitted updates and improves terminal metrics relative to baselines including uncalibrated VCG; in the common-fit Checker audit, mean Risk/Anchor is 0.4931 versus 0.8306 for LOO-selected ridge.

    Source: §1 · p. 2, §6 · p. 8

  • Takeaways & Limitations

    RCPT provides an inexpensive way to retain supported adaptation when fitting information is limited.

    Source: §1 · p. 2, §7 · p. 9

    The structured guarantee is conditional on calibration error, and residual upper bounds do not impose a universal ordering of endpoints or genealogy.

    Source: §6 · p. 9

Abstract

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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.

1 Introduction

Finite-particle VCG can improve its fitting residual while worsening deployment risk, so a fitted update must be judged by how well it generalizes. RCPT calibrates the retained update using existing regression information, and experiments show it mitigates harmful fits across molecular and image tasks.

1 Introduction

  • Feynman–Kac correction and ACE use importance weights and SMC to address proposal mismatch, but finite-particle resampling can deplete ancestral diversity while unweighted deployment retains proposal bias.Inference-time steering adapts pretrained diffusion or flow models by changing the dynamics transporting noise toward data.Source: §1 · p. 1
  • With limited fitting information, VCG can nearly eliminate its fitting residual while increasing independent residual risk by orders of magnitude.Increasing the fitting population costs additional expert evaluations.Source: §1 · p. 2
  • Expected out-of-fit improvement equals population headroom minus coefficient-estimation penalty, separating ideal improvement from estimation cost.The centered Feynman–Kac rate is the normalized transport residual, and residual analysis also bounds unweighted terminal error.Source: §1 · p. 2
  • RCPT uses deletion leave-one-out residuals to calibrate the retained fraction of a VCG update, with no additional model calls and small linear-algebra overhead.It supports frozen or online deployment and retains more of VCG as fitting information grows.Source: §1 · p. 2
  • Across molecular and image domains, RCPT mitigates harmful fitted updates and improves terminal metrics relative to baselines including uncalibrated VCG.The evidence includes scaffold and molecular SMC protocols and CIFAR-10 generation.Source: §1 · p. 2
  • Figure 1 compares fitting and independent risk as log-scaled ratios to Anchor; lower vertical values indicate better independent risk, and colors denote fitting population.The caption cautions that a smaller fitting residual alone is insufficient.Source: Figure 1 · p. 2

2 Related Work

Related work spans proposal construction and correction, fitted guidance methods, and regression regularization. RCPT is distinguished by calibrating the deployment risk of an estimated VCG coefficient rather than changing its fitted directions.

2 Related Work

  • Score composition and reward guidance construct Anchor proposals, while FKC and ACE correct prescribed paths through importance weighting and SMC.Finite-particle costs and ancestral collapse motivate improving proposals before correction.Source: §2 · p. 3
  • VCG minimizes Feynman–Kac rate variance, whereas ECG fits gradient corrections through an energy objective; RCPT addresses estimation error in VCG’s fitted coefficient.Related approaches include adaptive importance sampling, controlled or twisted SMC, and diffusion or incremental-divergence objectives.Source: §2 · p. 3
  • Ridge penalizes coefficients and PCR truncates fitted directions, while RCPT calibrates one update amplitude using structured headroom and penalty estimates.In the common-fit Checker audit, RCPT has lower mean reference risk than Direct LOO and LOO-selected ridge/PCR.Source: §2 · p. 3

3 Finite-Particle Proposal Risk

The paper frames finite-particle proposal fitting as a deployment-risk problem: estimation error can erase population improvement, while scaling the fitted correction offers a way to limit that risk. RCPT estimates the tradeoff, and residual risk connects proposal quality to transport error and correction-weight cost.

3 Finite-Particle Proposal Risk

  • The proposal-risk setup keeps the target path fixed while distinguishing unweighted Proposal-Only deployment from FKC/ACE deployment with incremental weights and resampling.Population VCG is a projection over an additive drift correction, with coefficient errors measured in the residual-variance geometry C rather than Euclidean size.Source: §3 · p. 3

3.1 Headroom and coefficient-estimation error

  • The exact out-of-fit risk decomposition defines population headroom and coefficient-estimation penalty, with the full estimated update harming expected residual risk precisely when the penalty exceeds headroom.The penalty includes coefficient variance and squared bias, so the identity does not require unbiased fitting.Source: §3.1 · p. 4

3.2 Scaling the fitted correction to minimize risk

  • When mean estimation error is unaligned with the population coefficient under C, the risk-minimizing fixed amplitude is H/(H + P), retaining more of the fit as headroom grows relative to estimation error.Unbiased fitting is sufficient for this zero-alignment condition, but is not required.Source: §3.2 · p. 4

4 Risk-Calibrated Proposal Transport

  • RCPT uses deletion leave-one-out residuals and a structured penalty estimate to select the retained VCG fraction, returning to Anchor when estimated penalty exhausts empirical gain.It reuses VCG feature values and divergences, adds no model calls, and incurs small dense deletion-refit systems.Source: §4 · p. 5

4.1 Residual risk upper-bounds Wasserstein error

  • Residual risk bounds unweighted transport error under stated regularity assumptions and controls leading local correction-weight variance for corrected sampling.The analysis retains the residual’s time profile before an optional Poincaré conversion to variance.Source: §4.1 · pp. 5–6

5 Experiments

The experiments test RCPT across Checker distributions, molecular generation, and CIFAR-10, alongside comparisons with alternative calibration and regularization choices. Across molecular and image domains, RCPT mitigates harmful fitted updates and improves terminal metrics relative to baselines including uncalibrated VCG.

5.1 Synthetic Checker Distribution

  • RCPT has the lowest mean reference risk at every fitting budget and lower mean W1 and W2 than Anchor, VCG, and ECG in both matched deployment modes.Under SMC, it also has the lowest MMD and retains 568.5 terminal lineages versus 338.7 for Anchor; Proposal-Only MMD favors Anchor.Source: Table 1 · p. 6, Table 2 · p. 6, §5.1 · p. 7

5.2 Scaffold Decoration on CrossDocked2020

  • Across 76 scaffold–pocket tasks, RCPT restores useful generation after expanded VCG fitting: validity rises from 0.229 to 0.434 and unique useful yield from 0.118 to 0.251.The paired 95% pocket-bootstrap improvement intervals are [0.151, 0.260] for validity and [0.086, 0.181] for unique useful yield.Source: §5.2 · p. 7, Table 3 · p. 7

5.3 Multi-Property Molecular Reward Steering on QM9

  • In split QM9 deployment, RCPT raises unique successful molecules from 13.88 to 23.00 per 128 outputs and surviving-lineage fraction from 0.1611 to 0.3525.Online, both VCG and RCPT preserve all 144 lineages; RCPT retains mean amplitude 0.945 and achieves slightly better mean evaluator error and unique success.Source: §5.3 · pp. 7–8, Table 4 · p. 8

5.4 Improving Guided Image Generation on CIFAR-10

  • On CIFAR-10, RCPT improves all six externally evaluated metrics over Anchor and VCG-9; versus VCG-9, independent accuracy rises from 0.6438 to 0.6750.KID falls from 0.2685 to 0.2407 and Clean-FID from 89.759 to 88.232; the comparison uses one seed.Source: Table 5 · p. 8, §5.4 · p. 8

6 Discussion

  • At N = 16, Checker mean Risk/Anchor is 0.4931 for RCPT, versus 0.8306 for LOO-selected ridge and 1.1740 for PCR.RCPT captures 80.7% of reference-ray improvement; reducing controller capacity can stabilize residual fitting yet sacrifice downstream chemistry.Source: §6 · p. 8
  • Calibration alternatives transfer unevenly: fixed α = 0.50 exceeds RCPT’s molecular yields on a ten-pocket scaffold subset but worsens mean QM9 evaluator error relative to VCG.Against Direct LOO, RCPT improves Checker risk at N = 16 and mean online QM9 unique success, while Scaffold downstream intervals include zero.Source: §6 · p. 9
  • Residual bounds do not universally order terminal metrics or genealogy, and extending frozen-schedule guarantees to online interacting particles requires dependence and target-reference controls.The common-fit N = 32 SMC audit favors Anchor over RCPT in W2 by 0.0817, with paired 95% interval [0.0051, 0.1629].Source: §6 · p. 9, A Supplementary Overview · p. 16

7 Conclusion

  • Across molecular and image domains, RCPT mitigates harmful fitted updates and improves terminal metrics relative to baselines including uncalibrated VCG.The conclusion describes controlled and molecular deployments as practical repairs of harmful VCG fits.Source: §7 · p. 9

B.1 Common-fit calibration and paired comparisons

The common-fit Checker comparison favors RCPT over Direct LOO, LOO-selected ridge, and LOO-selected PCR on paired reference-risk differences. The remaining subsections describe scaffold capacity and calibration cohorts, molecular and image audits, application-cost measurements, and analytic assumptions.

B.1 Common-fit calibration and paired comparisons

  • RCPT captured 46.8%, 80.7%, and 86.3% of reference-ray improvement at N = 8, 16, and 32, respectively.These are ratios of aggregate improvements, not means of per-fit fractions; the reference-ray oracle is a diagnostic using evaluation states.Source: §B.1 · p. 16, Table B.1 · p. 16
  • RCPT-minus-comparator Risk/Anchor differences were −0.6331 for Direct LOO, −0.3375 for LOO-selected ridge, and −0.6804 for LOO-selected PCR; all paired 95% intervals excluded zero.The comparison uses the same full-data VCG direction; Table B.1 also reports captured improvement and curvature-weighted ray regret.Source: Table B.1 · p. 16, §B.1 · p. 16

B.2 Scaffold capacity and calibration

  • In the 76-pocket scaffold capacity ablation, mean RCPT amplitude was 0.194 at fit 16 and 0.906 at fit 64.The cohort used 16 deployment slots per arm and a fixed generation denominator.Source: Table B.2 · p. 17
  • The separate fit-16/deploy-16 scaffold cohort compares Anchor, Direct LOO, RCPT, and full VCG using active-path risk; paired uncertainty is reported separately.Its means are distinct from the earlier capacity cohort.Source: Table B.3 · p. 17

B.3 QM9 and image calibration

  • The QM9 calibration ablation uses eight online seeds at N = 144, while the CIFAR-10 ablation compares Direct LOO and RCPT on the same 320-output seed and nine auxiliary fields.The QM9 table defines unique success as distinct valid target-meeting identities per 144 outputs; the CIFAR-10 table caption supplies no outcome values.Source: Table B.4 · p. 17, Table B.5 · p. 17

B.4 Application cost

  • The Checker selector took 0.132 seconds per 200-step schedule for RCPT, versus 1.31 for Direct LOO, 9.34 for selected ridge, and 13.03 for selected PCR.These implementation-specific selector timings are separate from hardware-identified generation measurements.Source: §B.4 · p. 17

C.1 Analytic setup

  • The analytic setup assumes a positive target with finite normalizer and sufficient regularity, a well-posed Markov evolution, and the stated risk bounds.The assumptions do not identify a particle-dependent online drift with a fixed single-particle Markov process.Source: §C.1 · p. 18

C.2 Proof of Theorem 4

The proof links the centered transport residual to target-path mismatch and derives terminal bounds for the unweighted proposal under stated assumptions. It also distinguishes residual control from finite-particle genealogy and endpoint guarantees.

C.2 Proof of Theorem 4

  • The normalized source identity yields a total-variation bound, but zero residual along the whole path—not merely a reduced residual bound—implies exact unweighted marginals.Propagated signed defects can cancel, so a smaller residual alone need not strictly reduce terminal discrepancy.Source: §C.2 · pp. 18–19

C.3 Proof of the Wasserstein upper bound

  • The minimum-energy correction is a gradient field, and a Poincaré constant bounds its energy by residual variance; the Wasserstein argument couples corrected and proposal processes with shared noise.The energy bound itself does not require Poincaré.Source: §C.3 · p. 19
  • The terminal Wasserstein bounds concern the unweighted proposal, not finite corrected-particle error, which also depends on weights, dependence, resampling, and mixing.The local weight result addresses a separate benefit of reducing residual variance.Source: §C.3 · p. 19

C.4 Proof of Proposition 1

  • Normalizing the local Feynman–Kac increment subtracts the target expectation of the rate at first order, under the stated local regularity and remainder conditions.Uniform conclusions require uniform remainder control.Source: §C.4 · p. 19

C.5 Why local variance does not order genealogy

  • Equal marginal incremental second moments do not determine cumulative weight variation or genealogy; resampling timing and cross-covariances also matter.The paper therefore reports surviving initial ancestors and lineage ESS, not numerical weights alone.Source: §C.5 · p. 20

C.6 Time discretization and approximate residuals

  • Discretization and approximation bounds require their stated smoothness and model-error controls; finite reference samples and interacting SMC outputs add separate errors.The interacting-particle error cannot be replaced by an iid sampling rate without justification.Source: §C.6 · p. 20

D.1 Population projection; D.2 Proof of Theorem 1

  • The population VCG projection gives an exact headroom–penalty identity: expected fitted residual risk is the optimum residual variance plus coefficient-estimation penalty.The projection argument also covers singular covariance matrices.Source: §D.1 · p. 21, §D.2 · p. 21

D.3 Proof of Theorem 2; D.4 Proof of Theorem 3; D.5 What the structured model assumes

  • The structured amplitude oracle depends on alignment bias, and population formulas do not transfer unchanged to estimated amplitudes or guarantee unbiased headroom estimation.Direct LOO need not dominate structured RCPT: lower amplitude-estimation variance can be preferable, and the common-fit audit distinguishes amplitude error from curvature-weighted ray regret.Source: §D.5 · pp. 21–22

D.6 Finite-particle penalty and its proof

The finite-particle penalty has a leading asymptotic form under fixed-dictionary, independent-sample regularity assumptions. The passages distinguish that result from fixed-ridge and dependent-particle settings and state a frozen-schedule consequence.

D.6 Finite-particle penalty and its proof

  • For iid sampling with a fixed dictionary, fixed-rank C, the stated moment and empirical-rank conditions, and the additional uniform-integrability condition, minimum-norm least squares gives P = N^-1 tr(ΩC†) + o(N^-1).The asymptotic lemma does not assert accuracy near interpolation.Source: §D.6 · p. 22
  • The centered CLT is not a fixed-ridge theorem, and random importance weights or interacting particles require their own covariance analysis.Conventional weight ESS does not generally estimate the regression-score covariance; equal weights after resampling do not imply independent observations.Source: §D.6 · p. 22
  • For independently fitted frozen schedules, the projection identity combines with the dynamic Wasserstein bound under Theorem 4 assumptions; online-loop guarantees require separate analysis.Interpolation, model error, and numerical integration also need separate controls.Source: §D.6 · p. 23

E Calibration Ablation and ECG Baseline

Direct LOO changes the amplitude-calibration criterion while retaining the full-data VCG direction. ECG fits a distinct energy objective, and directional shrinkage offers an oracle gain over a common scalar amplitude when coordinatewise benefits differ.

E.1 Direct LOO

  • Direct LOO estimates deleted prediction loss without a scalar covariance model and changes only calibration, retaining the full-data VCG direction.The deleted fits use a fixed unscaled intercept and consistently normalized deletion weights.Source: §E.1 · p. 23

E.2 Residual fitting versus energy fitting

  • ECG solves a weak Poisson energy objective for gradient controls, which need not select the same field as VCG’s residual-variance objective.Its scalar-potential class does not automatically include VCG’s arbitrary gated vector fields; this is a difference in representation and objective, not a general superiority result.Source: §E.2 · p. 23

F.1 The oracle benefit of directional shrinkage

  • The directional oracle uses coordinate-specific amplitudes and can achieve lower coefficient risk than a scalar oracle when coordinatewise headroom and penalty differ.Estimating additional amplitudes has a cost, so reduced rank and shrinkage are complementary decisions.Source: §F.1 · p. 24

F.2 An exact sample on which directional LOO is worse

The appendix separates the potential benefit of a richer oracle from the reliability of estimating it, then specifies conditions and implementation boundaries for applying calibration.

F.2 An exact sample on which directional LOO is worse

  • A positive-probability sample makes the two-coordinate directional criterion uniquely minimize at (1, 0), while the scalar criterion minimizes at 143/290.The counterexample concerns samplewise dominance, not risks averaged over fitting samples.Source: §F.2 · p. 24

F.3 When the bias term can be omitted

  • Zero bias follows under a correctly specified conditional linear model with full empirical rank and unregularized least squares, but not from projection orthogonality alone.A coefficient bias of order N^-1 can make the bias term as large as the estimation penalty; ridge contraction can also invalidate the zero-bias interpretation without sampling noise.Source: §F.3 · p. 24

G Specialization to FKC and ACE

  • RCPT does not repair an invalid or singular fixed-exponent path; ACE first repairs the schedule while preserving the desired endpoints.Proposal fitting then acts on that repaired path.Source: G Specialization to FKC and ACE · p. 25

H.1 Stable weighted least squares

  • Weighted least squares uses centered covariances and a float64 solve, while an empirical-variance guard can fall back to Anchor without guaranteeing out-of-fit safety.The deployed field after clipping or fallback must be distinguished from the theoretically analyzed scalar-selected field.Source: §H.1 · p. 25

H.2 Basis and divergence by domain

  • The image setup uses forward-noised training images for fitting and independent test images for held-out states, whereas molecular setups use fixed controller bases for decoration and reward-tilted optimization.For images, common-state risks do not directly evaluate the exact target-path expectation.Source: §H.2 · p. 26

H.3 Deletion scope and computational complexity

  • After field values and divergences are available, the VCG solve costs O(Nm^2 + m^3), and RCPT adds dense leave-one-out algebra without an expert or denoiser call.Hardware-specific timing and particle scaling are reported separately.Source: §H.3 · p. 26

H.4 Evaluation laws and dependent particles

The evaluation law limits what the audits establish: common-state and corrected-SMC references are not automatically target-law risks, and Checker endpoint accuracy can rank differently from particle genealogy.

H.4 Evaluation laws and dependent particles

  • Theoretical risk is defined under the target marginal, but common-state evaluation and active-path evaluation do not automatically estimate that target-law expectation.The Checker corrected-SMC reference is approximate, and the molecular common-state audit does not quantify its gap to the target law.Source: §H.4 · p. 27
  • Equal weights after resampling do not establish independent fitting information because descendants can share ancestors, so weight ESS alone does not justify an online guarantee.Lineage-aware deletion and particle-system concentration require separate analysis [11].Source: §H.4 · p. 27

I.1 Controlled targets

  • The Checker comparison uses a schedule fitted on 16 particles and frozen before 2,000 fresh Proposal-Only or SMC trajectories, with both modes sharing the target and evaluation setup.The residual diagnostic is an active-path integral, not an independent target-law risk estimate.Source: §I.1 · p. 27
  • In the Checker calibration ablation, RCPT has lower mean W1 and W2 than Direct LOO in both deployment modes and lower mean MMD under SMC, while Direct LOO retains more SMC ancestors on average.Direct LOO retains 630.7 versus RCPT's 568.5 SMC ancestors; endpoint accuracy and genealogy have distinct rankings.Source: §I.1 · p. 28

I.2 Common-fit realized-ray and adaptive-regularization audit

The audit finds lower reference risk for RCPT among the implemented rules, but terminal transport rankings and calibration diagnostics are not uniformly aligned with that result.

I.2 Common-fit realized-ray and adaptive-regularization audit

  • The independent corrected-SMC reference uses two 16,384-particle systems and is neither exact target integration nor iid target sampling.Agreement between the systems measures simulation sensitivity, not shared model bias or the target-reference covariance gap.Source: §I.2 · p. 28
  • RCPT's mean integrated ray regret is 8.359, 4.500, and 3.776 across increasing N, versus Direct LOO's 12.900, 27.960, and 4.635, although Direct LOO has smaller mean absolute amplitude error at N = 8 and 32.Curvature weights amplitude errors differently across times, so amplitude accuracy and decision quality need not rank alike.Source: §I.2 · p. 29
  • At all three fitting populations, RCPT has the lowest mean reference risk among the implemented rules; at N = 16, its Risk/Anchor differences versus Direct LOO, ridge, and PCR are −0.6331, −0.3375, and −0.6804.At N = 8 and 32, intervals versus Direct LOO and ridge include zero, while intervals versus PCR remain below zero; these are pairwise intervals over ten fits, not simultaneous guarantees.Source: Table I.3 · p. 29, §I.2 · p. 29
  • The oracle gain over the better endpoint exceeds two-system sensitivity in 22/63 population-size–time cells, but the audit provides no finite-sample certificate for the implemented selector.The plug-in error bound additionally requires control of realized approximation errors and target-reference transfer.Source: §I.2 · p. 29
  • RCPT has lower mean Proposal-Only W2 than Direct LOO, ridge, and PCR at each budget, but SMC favors ridge at N = 16 and ridge and PCR at N = 32.At N = 32, the paired RCPT-minus-Anchor SMC W2 difference is +0.0817, with 95% bootstrap interval [+0.0051, +0.1629]; the transport bounds do not order these observed metrics.Source: §I.2 · p. 29
  • Mean selector time per 200-step schedule is 0.132 seconds for RCPT, versus 1.31 for Direct LOO, 9.34 for ridge selection, and 13.03 for PCR.These are selector timings within this implementation, not total sampling times or hardware-independent complexity comparisons.Source: §I.2 · p. 29

I.3 Scaffold protocol

Across scaffold, molecular, image, and diagnostic studies, RCPT often repairs harmful fitted proposals, with results that vary by task and evaluation metric. The audits also examine when calibration is reliable and how its trust changes with fitting information.

I.4 QM9 steering protocol

  • Both VCG and RCPT preserved all 144 lineages online; RCPT retained mean amplitude 0.945 and achieved slightly better mean evaluator error and unique success.The online protocol refits the live 144-particle population, whose particles continue and become outputs.Source: §I.4 · p. 31

I.5 CIFAR-10 class-conditional image generation

  • RCPT improved six of seven image metrics over matched Anchor and tied DLSM accuracy, but its single-seed comparison does not estimate between-fit uncertainty.Independent accuracy rose from 0.6625 to 0.6750, KID fell from 0.2688 to 0.2407, and Clean-FID fell from 89.043 to 88.232.Source: §I.5 · p. 34

I.6 Ten-pocket scaffold fixed-rule comparison

  • On the matched ten-pocket subset, RCPT raised validity from 0.319 to 0.563 and scaffold yield from 0.319 to 0.513 versus VCG, but fixed α = 0.50 led on several yields.The result supports adaptive negative-transfer protection, not dominance over an oracle-chosen fixed scalar for every subset.Source: §I.6 · p. 35, Table I.11 · p. 35

I.7 QM9 steering fixed-rule comparison

  • No fixed scalar improved mean MAE over VCG, whereas RCPT lowered MAE from 0.8499 to 0.8252 and raised unique success from 33.5 to 39.2.With four seeds, all paired MAE intervals against VCG included zero, so this audit does not establish statistically decisive reward-domain superiority.Source: §I.7 · pp. 35–36

I.8 Bias and penalty-estimator audits

  • Ignoring coefficient bias increased integrated oracle risk by 0.781%; the structured penalty proxy correlated 0.893 with N^-1 tr(ΩC†) but understated its integrated scale.A separate realized-calibration audit found lower selected-ray regret despite substantial proxy error, but its approximate dependent reference does not certify the target-law bound.Source: §I.8 · p. 36

J.1 Mechanism diagnostics

  • At fit 8, median residual risks were 105.22 for VCG and 0.966 for RCPT; means were 524.17 and 1.084, reflecting a remaining RCPT tail.RCPT improved 66/70 Checker runs and repaired 34/47 harmful VCG runs to Anchor or below.Source: §J.1 · pp. 36–37

J Additional Mechanism Diagnostics

  • RCPT’s mean amplitude rose from 0.143 at N = 4 to 0.921 at N = 256 as VCG and RCPT independent risks converged.Amplitudes describe the rule’s response in each plotted cohort, not a comparable drift norm or a terminal-optimal amplitude.Source: §J.1 · p. 37

J.2 Selected amplitude across tasks

Across tasks, RCPT’s selected amplitude varies by cohort, and fixed heuristics do not transfer uniformly. Held-out residual risk tracks terminal error comparatively, but is not an exact terminal-error estimate.

J.2 Selected amplitude across tasks

  • Across the plotted cohorts, Checker amplitudes were strong early then smaller, while CIFAR amplitudes were mostly conservative; these curves do not identify terminal-optimal amplitudes.Amplitudes are relative to fitted coefficients, not comparable drift norms or independent measurements of population headroom.Source: §J.2 · p. 37

J.3 Residual risk as a terminal surrogate

  • Integrated held-out residual correlated with empirical TV at Spearman 0.775, while theorem-aligned B1 correlated at 0.730 and was a median 47% tighter than the additional bound.Every evaluated B1 exceeded one, with median 5.99× empirical TV; residual risk is therefore a comparative diagnostic, not an exact terminal-error estimate.Source: §J.3 · p. 38

J.4 Cross-domain portability and fixed controls

  • The best fixed heuristic did not transfer across tasks: a fixed scalar was strong on selected scaffold yields, while none improved QM9 steering MAE over VCG.Ridge behavior was also task-dependent and nonmonotone; RCPT estimates trust from the available fit rather than selecting a task-specific regularization value.Source: §J.4 · p. 38

K Computational Cost

RCPT adds little computation to VCG in the reported implementations, and its main systems benefit is fitting a controller on the small ensemble intended for generation. Particle scaling measurements show that trajectory and field/model computation dominate controller-solve time.

K Computational Cost

  • RCPT added 3.13 seconds over VCG in approximately 3,880 seconds, or 0.081%, with equal probed memory.QM9 timings came from four A100-SXM4 runs; peak memory was measured in a matched 10-step probe.Source: K Computational Cost · p. 38
  • The systems benefit is fitting on the small ensemble users intend to generate, not making one trajectory cheaper through leave-one-out refitting.The measured 12-field fits from 8–16 particles add negligible computation over VCG, while larger fitting populations increase model cost and memory.Source: K Computational Cost · p. 38
  • From 16 to 64 scaffold particles, allocated VRAM rose 3.93× and wall time 2.12×, while aggregate controller-solve time changed by only 0.20 seconds.On the same A100, QM9 probes required 4.569 GB at N = 16 and 40.204 GB at N = 144; these are measurements, not extrapolations to thousands of particles.Source: K Computational Cost · p. 38, Table K.1 · p. 39
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