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When to Ask for Help: A Decision-Theoretic Account and Measurement Standard for Slow-Module Scheduling in Dual-System Vision-Language-Action Models

Shilin Zhang

arXiv:2610.21160v1cs.RO

TL;DR

Existing slow-module triggers lack a shared objective, measured costs, and a standard for separating schedule quality from module quality. This paper formalizes invocation as a calibrated decision problem and finds that adaptive gains depend on collapse heterogeneity, while measured value functions can be compiled into refresh periods close to the closed-loop oracle.

  • Problem

    Published slow-module triggers lack a shared objective, measured costs, and an evaluation standard separating schedule quality from slow-module quality.

    Source: §1 · p. 2

  • Method

    The paper frames scheduling as a decision problem in which adaptive gains depend on how much the channel reveals about collapse, and measures query cost and latent drift on FiS-VLA.

    Source: §2 · p. 6, §3.4 · p. 8

  • Results

    Calibrated advantage vanishes under homogeneous collapse and grows with collapse heterogeneity; compiled refresh periods track the closed-loop oracle with a cost gap of at most 2.4%.

    Source: §1 · pp. 3–4

  • Takeaways & Limitations

    Adaptive scheduling gains are tied to information about collapse, and the value-function compiler can produce per-task refresh periods close to the closed-loop oracle.

    Source: §1 · p. 3, §3.4 · p. 8

    The baseline is deliberately minimal, and its persistence assumption is optimistic for self-correcting collapse, which reduces the value of querying.

    Source: §3.4 · p. 8

Abstract

from arXiv · show

Dual-system vision-language-action (VLA) models pair a high-frequency fast policy with a slow, expensive module, and published trigger mechanisms for invoking it lack a shared objective, measured costs, and any standard separating schedule quality from module quality. We provide the missing account: invocation is formalized as a calibrated decision problem whose fixed-ratio, fixed-budget, and history-gated schedules are degenerate cases of one quantity, the Value of Intervention (VoI); a renewal baseline proves the optimal schedule periodic under state-independent collapse, with adaptive gains governed by the cost ratio $\rho$ and the dispersion of collapse. Measured systems span three orders of magnitude in $\rho$ (FiS-VLA $0.95$; RoboICL about $10^3$), and a direct test confirms the low-$\rho$ prediction: calibrated gains vanish under homogeneous collapse and grow with heterogeneity. Fourteen closed-loop arms on LIBERO test the contested middle of the continuum ($\rho=1.53$): the adaptive trigger allocates its queries exactly as the threshold rule predicts, yet the language-rewrite channel cannot convert them into success on either suite, and a revert actuator that undoes a failed rewrite is the only intervention with zero net downside on both LIBERO-Spatial and LIBERO-Goal, where the rewrite deficit deepens from thirteen to forty-two points. A four-measure leak-free protocol cuts apparent early-warning AUC from $0.93$ to $0.62$ on $929$ deployment traces, a detection-limited frontier theorem converts detection quality into a schedulability statement, and a compilation loop turns measured value functions into per-task refresh periods within $2.4\%$ of the closed-loop oracle. Together these results turn slow-module invocation from a design opinion into a measured decision.

1 Introduction

The paper frames slow-module invocation as a calibrated decision problem, unifying existing schedules through Value of Intervention and measuring when adaptive invocation can pay. Its introduction also reports that correct query allocation does not guarantee success when the slow module’s language channel cannot convert queries into useful actions.

1 Introduction

  • Value of Intervention unifies fixed-ratio, fixed-budget, and history-gated schedules; the renewal baseline predicts periodic scheduling under state-independent collapse.The optimal interval is 2cslow/(wfailπ); fixed ratios are hard to beat at low ρ and low collapse heterogeneity, while adaptivity pays as either grows.Source: §1 · p. 2
  • The calibrated router queries only when estimated Value of Intervention exceeds query cost; its advantage vanishes when the estimator loses validity on most steps.The reported advantage is insensitive to estimator variance, but requires the estimator to remain valid on most steps.Source: §1 · p. 2
  • On 929 RoboICL deployment traces, early-warning AUC falls from 0.93 post hoc to 0.62 leak-free, showing why the measurement protocol must prevent leakage.The paper reports that the leak-free AUC is 0.62 (p = 0.010).Source: §1 · p. 3
  • On controlled task families, compiled per-task refresh periods track the closed-loop oracle with a cost gap of at most 2.4%.The compiler turns a measured staleness value function into a refresh period; on the FiS backbone, it selects the boundary dictated by a nonpositive value function.Source: §1 · p. 3
  • At ρ = 1.53, 14 LIBERO arms show the trigger matches its threshold allocation, but rewrites fail to improve success; revert alone avoids net downside on both suites.The rewrite deficit deepens from thirteen to forty-two points on LIBERO-Goal, while the revert arm tracks fast-only there.Source: §1 · p. 3
  • On the testbed, calibrated advantage vanishes under homogeneous collapse and grows monotonically with collapse heterogeneity, as the renewal baseline predicts.The paper also reports that fine-tuning the open checkpoint on four consumer GPUs directly confirms transfer of the low-ρ prediction to deployed FiS.Source: §1 · p. 4

2 Related Work: Three Families, One Design Space

Prior dual-system VLA work includes fixed latent refreshes, value-guided candidate selection, and history-based reuse gates, while concurrent work adds uncertainty-triggered invocation. The paper relates these approaches to status-update and caching theory, positioning its contribution as a decision account with measured costs and a leak-free comparison standard.

Three Families, One Design Space

  • Fast-in-Slow fixes latent refreshes at a 1:4 ratio selected by ablation, whereas Hume fixes its candidate budget N and uses a trained value head to select actions, not schedule consultation.Fast-in-Slow’s reported 21.9 Hz is the whole-system control rate, yielding an effective slow refresh tick of about 5.5 Hz.Source: §2 · pp. 4–5
  • RoboICL’s history-based Jev gate reuses predicted action suffixes to cut calls by 33–48%, but is auxiliary to its main benchmarks and degrades some benchmarks.The paper analyzes the gate’s uneven task success as a missing-calibration failure mode.Source: §2 · p. 5
  • Concurrent triggers use decision tokens, action-reprediction dispersion, or low confidence; the paper treats them as estimators of collapse probability, not a shared account of when triggering can pay.The cited mechanisms do not report measured query costs, collapse heterogeneity, or leak-free evaluation.Source: §2 · p. 5
  • Status-update theory supplies threshold and square-root cost-to-drift results for scalar signals, while this paper transports and measures the Bernoulli-collapse instance on an embodied system.On FiS-VLA, it measures both query cost and latent drift rather than assuming them.Source: §2 · pp. 5–6
  • Caching theory prices staleness through age-dependent miss penalties, but the paper identifies multi-second VLM calls as a cost regime far beyond those literatures’ original focus.The paper positions its question as the budget for consulting a separate slow module inside a dual-system VLA, rather than budget allocation within one model.Source: §2 · p. 6
  • The paper distinguishes its account from prior trigger mechanisms by combining a shared objective, measured cost economics, and a leak-free standard for cross-system comparison.Its positioning also distinguishes scheduling consultation from fixed latent-refresh, candidate-selection, and reuse-gating mechanisms.Source: §2 · pp. 4, 6

3 Method: Routing as a Calibrated Decision Problem

This method frames slow-module invocation as a calibrated decision—query when intervention value exceeds cost—while treating fixed schedules as special cases and separating gains from collapse detection from value that survives the interface. It specifies a leak-free logistic-regression proxy and a two-head online router, not yet integrated with the deployed checkpoint, under a shared measurement contract for open or closed fast/slow model pairs.

3.1 Setup

  • A schedule chooses whether to query at each step; the optimal rule queries when the Value of Intervention exceeds the slow-query cost.The Value of Intervention is defined by the gap between slow-corrected and fast-only task value when that gap exceeds a margin.Source: §3.2 · p. 6, §3.1 · p. 6

3.3 Three Baselines as Degenerate Cases

  • Fixed-ratio scheduling assumes time-invariant collapse, fixed-budget scheduling assumes pervasive collapse, and history-only gates are suboptimal when collapse depends on state.The paper identifies history-only gates as inheriting variance that calibrated state estimates could remove when scheduling history carries no early signal of collapse.Source: §3.3 · p. 7

3.4 The Cost-Asymmetry Continuum

  • At ρ = 0.95, FiS-VLA’s measured low-asymmetry regime makes fixed cadence reasonable; adaptive gains vanish under time-invariant collapse and grow with its temporal dispersion.The periodic baseline is optimal under state-independent collapse; the minimal persistence model is optimistic for self-correcting collapse, while continuous-drift and event-collapse models share the same optimal-period scaling.Source: §3.4 · pp. 7–8
  • With informative between-query signals, optimal routing thresholds posterior collapse probability; periodic scheduling is optimal only when the signal carries no evidence about future collapse.The adaptivity gain depends on the channel’s information about collapse and vanishes with that information; a windowed trigger trades earlier queries and more false queries against later queries and fewer false queries.Source: §3.4 · p. 8
  • The interface scales or reverses the trigger boundary: routing queries when ηC VoI exceeds cost, while interface risk also depends on whether interventions stay within validated behavior support.Detection and channel value must be measured separately; an aggregate comparison can be null even when detection is effective, and a mean-null interface can still have substantial downside.Source: §3.4 · pp. 10–11
  • For a leak-free risk score, label-blind policies satisfy m = r, while AUC determines the binormal detection-limited envelope above that periodic-scheduling diagonal.Frontier certification must account for episode clustering: RoboICL traces have κ ≈ 0.91–0.99 and design effects of 8–18, reducing effective sample sizes below nominal decision counts.Source: §3.4 · pp. 11–12
  • Without a calibrated router or measured value function, querying once accrued staleness loss reaches query cost is 2-competitive; randomization improves the bound to e/(e−1).The measured linear drift law has slope h = 0.12 wfail, making the stated accrual assumption quantitative for this system.Source: §3.4 · pp. 12–13

3.5 The Calibrated Router

  • The calibrated-uncertainty router queries when estimated per-step collapse propensity exceeds a threshold, with threshold sweeps tracing the frontier and estimator quality varied through noise and rank corruption.Comparisons match query budgets; estimator quality is controlled by additive noise variance and rank-destroying corruption on a fraction of steps.Source: §3.5 · p. 13
  • A logistic regression over leak-free decision-history features is evaluated with episode-grouped cross-validation, preventing episodes from contributing to their own training fold.Features include dispatch mean, dispersion, short-horizon fraction, density, request–turn rejection rate, and last dispatch length.Source: §3.5 · p. 13
  • The proposed online router is a two-head multilayer perceptron with about 42k parameters and sub-millisecond inference, but is validated only through proxies (i) and (ii).One head predicts collapse probability using binary cross-entropy plus a calibration term; the other predicts measured quality gaps using squared error. Training touches only rθ, with πf and Ms frozen; integration on the deployed FiS checkpoint remains outstanding.Source: §3.5 · p. 13
  • A narrow reset–observe–propose–correct interface lets open fast/slow pairs and closed models share the same contract, with response snapshots frozen during measurement runs.Source: §3.5 · p. 13

4 The Four-Measure Protocol

The four-measure protocol separates schedule quality from slow-module quality using logged rollouts and query decisions. A five-question checklist adds controls for competence, leakage, censoring, measured costs, and frontier reporting.

4 The Four-Measure Protocol

  • The four measures are staleness curves, routing frontiers, failure attribution, and calibration reliability, computed from rollouts and query decisions.They expose when fast-only stretches fail, compare success against query rate, locate queries and failure onsets, and assess collapse-predictor reliability.Source: §4 · p. 14
  • AUC fell from 0.93 post hoc to 0.62 leak-free on deployment traces, showing why predictors must use only decision-time information.The protocol also reports reliability diagrams and separates whole-episode statistics as post-hoc descriptors, not predictive claims.Source: §4 · p. 14
  • The protocol is intended as an evaluation instrument, combining the four measures with competence, cost, and frontier checks before scheduling claims.The frontier compares a swept trade-off curve against the periodic diagonal and the Proposition 7 envelope.Source: §4 · p. 14, §4.1 · p. 15

4.1 A Five-Item Checklist

  • The five-item checklist asks whether competence permits meaningful comparisons, features are leak-free, censoring is controlled, costs use end-to-end wall-clock, and results show frontiers.Matched-prefix scoring is the minimal control for schedule-induced episode-length confounding; a single operating point cannot establish an advantage beyond the frontier envelope.Source: §4.1 · pp. 14–15
  • At 30 episodes, exact intervals require roughly a 0.25 success gap to separate two arms; detecting a 0.1 effect needs about 200 episodes per arm.Paired seeds and initial states reduce variance through discordant-pair counts, while Holm correction controls multiplicity across arm contrasts.Source: §4.1 · p. 15

5 Experiments

Experiments validate calibrated routing on a synthetic testbed, deployment traces, hardware, and controlled task families. In the testbed, routing retains an advantage as estimator noise rises, provided comparisons match query budgets.

5 Experiments

  • Across all 929 RoboICL traces, failed episodes query 25% less densely and have greater staleness, with category-dependent effects.The experiments also measure FiS-VLA refresh latency and test low-cost-asymmetry predictions, a closed-loop integration, and schedule compilation.Source: §5 · p. 15

5.1 Synthetic Testbed: Calibrated Routing Dominates Fixed Schedules

  • The synthetic testbed models bursty, state-dependent collapse and compares always-query, fixed-ratio, reuse-gate, and calibrated schedules across thresholds.Success averages five seeds, with 95% confidence intervals within ±0.003.Source: §5.1 · p. 15
  • At matched budget 0.25, calibrated routing beats the FiS-like fixed ratio by 6.4 points; at query rate 0.202, it dominates the Jev-style gate.At half the budget (0.127), it still matches the full-budget fixed 1:4 schedule.Source: §5.1 · pp. 15–16, Table 2 · p. 16

5.2 How Much Signal Quality Does Calibration Need?

  • The calibrated advantage declines from +0.049 at σu = 0.05 to +0.013 at σu = 0.90 without crossing zero under budget-matched noise.The comparison holds realized query rate near 0.25; additive noise preserves rank correlation, so it does not test systematic miscalibration.Source: §5.2 · p. 16

5.3 Where Calibration Actually Breaks

The key failure is not estimator variance but lost ranking validity: calibrated routing gains disappear when uncertainty is uninformative on most steps. When signal quality is retained, its advantage over fixed schedules persists, while baselines that ignore the signal cannot exploit it.

Where Calibration Actually Breaks

  • Across the budget range, the calibrated routing frontier sits below-and-left of the fixed-ratio frontier.Figure 2 plots success against query rate, with means and confidence intervals over five seeds.Source: Figure 2 · p. 17
  • The calibrated advantage is statistically indistinguishable from zero from p ≈0.8 onward and reaches the random-query limit at p = 1.The estimate is replaced by an independent uniform draw with probability p; benefit is seriously eroded when roughly half or more of steps lose rank information.Source: §5.3 · p. 17
  • Fixed-ratio, always-on, and reactive gates are invariant to signal quality because they never consult the uncertainty estimate.The calibrated router’s sensitivity is the cost of using the signal, with the crossover near p ≈0.8 marking when that cost exceeds the benefit.Source: §5.3 · p. 17
  • At matched query budgets, the advantage of calibrated routing over the fixed 1:4 schedule decays monotonically as uncertainty-signal quality worsens, but remains positive.Figure 3 reports budget-matched success against signal quality and the calibrated advantage over the fixed schedule.Source: Figure 3 · p. 18
  • Failed RoboICL episodes query less densely and run with greater staleness; the per-episode staleness gap is +2.5 steps (p = 0.0005).The gap is significant under label permutation.Source: Table 3 · p. 18

5.4 Deployment Traces at the High-Asymmetry End

Across 929 RoboICL evaluation episodes, dispatch-history features provide only modest leak-free early-warning signal, and that signal weakens with longer prefixes. The controls show that whole-episode prediction is inflated by episode length and censoring.

Deployment Traces at the High-Asymmetry End

  • The trace set contains 929 evaluation episodes across four task categories, with decision-level records at 25 Hz.Each decision records its step and the number of fast steps dispatched since the last slow query.Source: §5.4 · p. 18
  • Figure 4 shows early-warning AUC by prefix length K under four settings, with a permuted-label null of 0.50 and significance vanishing as the prefix grows.The caption’s stated decline motivates interpreting early-prefix results separately from longer-prefix scores.Source: Figure 4 · p. 19
  • AUC falls from 0.933 post hoc to 0.62 leak-free for the first K ≤10 decisions, with p = 0.010 against the permuted-label null.Whole-episode AUC is dominated by episode length, while the early result uses dispatch statistics only and does not require length or density features.Source: §5.4 · p. 19
  • Under censoring-matched evaluation, AUC declines from 0.62 to 0.59 at K=10 and reaches 0.47 at K=20 (p = 0.81).The comparison includes only episodes long enough for the prefix to cover at most half the trajectory.Source: §5.4 · p. 19
  • Early dispatch irregularity forecasts failure weakly and only in the first few decisions, so schedule features are useful to include but unsafe to rely on alone.The paper recommends fusing them with online state uncertainty; without leak-free and censoring controls, the early-warning claim would appear roughly 1.5× stronger.Source: §5.4 · p. 19

5.5 Decision-Level Routing Frontier

At the decision level, history features yield a usable but imperfect routing frontier, while episode clustering materially widens uncertainty around its operating points. The measured cost context places RoboICL at very high asymmetry, where missed failures—not call cost—bind reuse.

Decision-Level Routing Frontier

  • Leak-free per-decision AUC is 0.646 over the first ten decisions and 0.636 over the first twenty, with category values of Open 0.74, Long-Horizon 0.54, Precision 0.52, and Memory 0.45.Memory is below chance, and category structure sharpens the aggregate reversal.Source: §5.5 · p. 20
  • Matched-cohort AUC holds at 0.649 for a ten-decision prefix but falls to 0.592 for twenty decisions, confirming that longer-prefix signal is partly censoring-induced.The router should be calibrated against the matched numbers.Source: §5.5 · p. 20
  • At r = 0.38, the router misses m = 0.28 versus the periodic policy’s m = r = 0.38; at m ≤0.20, it sustains r up to 0.33.The empirical curve lies below the diagonal, and its binormal envelope sits 5–20% above the empirical curve across the operating range.Source: §5.5 · p. 20
  • Episode clustering expands the headline interval at r ≈0.39, m = 0.28 from naive ±0.007 to design-effect ±0.030, matching the block-bootstrap ±0.029.Measured intraclass correlations are κ = 0.91–0.99 and design effects are 7.6–17.6, making effective sample size about one thousand rather than 16,449.Source: §5.5 · p. 20
  • Table 4 reports FiS-VLA timing on one RTX 4090; one slow refresh costs about one fast action, or 3.8 denoising steps.The caption reports 10–12 timed runs after warmup and the observed run-to-run timing spreads.Source: Table 4 · p. 21
  • At ρ ≈10^3, even one-segment reuse pays for itself, making missed-failure rate m rather than call cost the binding constraint.The reported Jev gate drop on Align Blocks is 3/5 to 2/5.Source: §5.5 · p. 21

5.6 Cost Instantiation: the Low-Asymmetry End, Measured

The paper measures cost asymmetry using deployment-path wall-clock times and places FiS-VLA near the low-asymmetry end of the continuum. Its periodic baseline is already efficient there, so any adaptive frontier shift must come from collapse heterogeneity; this section measures the cost side only.

Cost Instantiation: the Low-Asymmetry End, Measured

  • FiS-VLA timing uses the public 7B checkpoint on one RTX 4090, with a slow query defined as one slow-system forward call and a fast action as one four-step DDIM forward call.The setup discloses that classifier-free guidance is unavailable and the fast path requires an in-memory KV-cache indexing fix.Source: §5.6 · p. 21
  • The operational ratio ρ = tslow/tfast uses end-to-end wall-clock time, including transport and API queueing for slow calls, and the deployed fast-loop step period.With weights open, the ratio is measured on the deployment path; with weights closed, it is read from reported latencies.Source: §5.6 · p. 21
  • On the same RTX 4090, OpenVLA-7B’s LIBERO-Spatial step is 299.6 ms median, while Qwen2.5-VL-7B-Instruct takes 458 ms for a one-sentence subgoal (ρ = 1.53) or 2.15 s for a 78-token grounded plan (ρ = 7.19).Both modules run on the same card, excluding network transport and queueing from the ratio.Source: §5.6 · p. 21
  • Under Table 5’s one-protocol cost-asymmetry continuum, the empirical span is three orders of magnitude; in LIBERO, co-locating both modules on one GPU moves a separate-VLM system an order of magnitude down the axis.Table 5 combines this paper’s measured runs with published latency and control-rate reports.Source: Table 5 · p. 22
  • At ρ ≈1, the periodic baseline is already efficient: the fixed 1:4 cadence spends +24% compute, and adaptive-trigger headroom is bounded by collapse heterogeneity.A refresh is worth 3.8 fast steps in denoising-step units.Source: §5.6 · p. 22
  • The measured cost side does not establish whether calibrated routing shifts FiS-VLA’s frontier at equal task success; at ρ ≈1, any shift must come from collapse heterogeneity.The paper frames the integration result as a testable prediction, with a null or small result calibrating rather than falsifying the continuum.Source: §5.6 · p. 22

5.7 Testing the Continuum Prediction on the Testbed

At fixed mean collapse and cost ratio ρ = 0.95, calibrated routing’s advantage over the fixed 1:4 schedule vanished under homogeneous collapse and grew with collapse heterogeneity, as the renewal prediction states.

5.7 Testing the Continuum Prediction on the Testbed

  • The test held time-averaged collapse at the E0-v2 level and varied the hard-step fraction w from 0 to 0.25, matching budgets against a fixed 1:4 schedule.At ρ = 0.95, the fixed schedule spent q = 0.250, a +23.7% compute overhead, matching the deployed cadence’s +24%.Source: §5.7 · p. 22
  • +0.059 ± 0.005 was the calibrated-routing advantage at the E0-v2 heterogeneity level, versus +0.005 ± 0.004 at homogeneous collapse.The homogeneous result was statistically indistinguishable from zero; the small residual reflected budget matching, with the closest threshold overspending by 7.6%.Source: §5.7 · p. 22, Table 6 · p. 23
  • At ρ ≈ 1, adaptive routing’s headroom over a well-tuned fixed ratio comes from collapse heterogeneity; homogeneous collapse leaves no threshold that beats it.This is the measured low-ρ prediction of the renewal baseline.Source: §5.7 · p. 23, Table 6 · p. 23

5.8 The Measured Frontier on the Fine-Tuned Checkpoint

On the fine-tuned FiS-VLA checkpoint, staleness had no positive fidelity cost over the measured age range, so the sparsest measured refresh schedule reduced latency without sacrificing action fidelity. Closed-loop RLBench runs failed because of policy competence, making them uninformative for scheduling comparisons.

5.8 The Measured Frontier on the Fine-Tuned Checkpoint

  • ∆ was nonpositive at every measured age: −0.02, −0.11, −0.16, and −0.25 MSE for δ = 1, 2, 4, and 8 fast steps.The tested range therefore showed no fidelity cost from staleness.Source: §5.8 · p. 23
  • Latent displacement rose from 0.20 to 0.90 across the measured sequence; replacing it with one from a different task’s episode raised action MSE by +0.275 (+11%).The action map was locally insensitive to this drift at the measured scale, although the latent carried task-relevant information.Source: §5.8 · p. 23
  • 107.6 ms at k=1 fell to 50.8 ms at k=8, a 2.1× latency reduction at equal or better action fidelity.The measured frontier used bf16 inference on one RTX 4090; its table reports latency amortized from cslow = 65.3 ms and cfast = 42.5 ms.Source: Table 7 · p. 24, §5.8 · p. 24
  • Refreshing more often than once every eight steps bought nothing in-distribution, and the measured frontier favored the sparsest tested schedule.Because ∆(δ) ≤ 0 over the measured range, the periodic baseline had no interior optimum there.Source: §5.8 · p. 24
  • Neither head-only nor LoRA fine-tuning produced a successful closed-loop RLBench run; both failed when the planner rejected unreachable targets, so the competence floor made schedule comparisons uninformative.The authors attribute the failures to compounded action error, not staleness; a success-level frontier requires a full fine-tune that the 96 GB budget could not fit.Source: §5.8 · pp. 23–24

5.9 Compiling the Measured Value Function into Schedules

The compiler measures a task’s staleness value function, minimizes renewal cost over candidate periods, and validates the resulting schedule against a closed-loop oracle. Its controlled-family schedules tracked the oracle closely, while measured nonpositive staleness on the real checkpoint sent tasks to the boundary of the probed range.

5.9 Compiling the Measured Value Function into Schedules

  • The loop measures ∆(δ), minimizes C(k) = cslow/k + cfast + wfail ∆(k) over periods, then validates the compiled period against the closed-loop oracle.Validation must use the same accounting: latent refresh carries the slow module’s benefit, and latent age resets only on queries.Source: §5.9 · p. 24
  • Unconditional rescue of failed steps made queries appear worthless and pushed every oracle period to the boundary, so the authors fixed this accounting artifact before reporting results.The compiler’s benefit is represented by latent refresh inside ∆, with no separate rescue term.Source: §5.9 · p. 25
  • 143 of 144 optimality certificates were valid against synthetic-family ground truth, and certified gaps reached 0.002–0.004 cost units at N = 20,000.The certificates combined an independent-sample upper bound with a lower bound averaged over independent probe batches.Source: §5.9 · p. 25
  • At N = 2,000 validation episodes, gap-proportional allocation raised oracle-period recovery from 0.33–0.61 under uniform allocation to 0.67–0.93.The near-tie cluster absorbed 95–98% of the budget; at N = 200, neither allocation separated the optimum.Source: §5.9 · p. 25
  • At most 2.4% was the compiled schedule’s cost gap from the closed-loop oracle across controlled task families and probe budgets.At 20,000 probe steps per age, compilation recovered interior optima when present and stayed at the boundary when staleness was flat; the real FiS row also compiled to the probed-range edge.Source: §5.9 · p. 25, Table 8 · p. 26

5.10 A Closed-Loop Test at the Middle of the Continuum

At ρ = 1.53, the LIBERO arms varied slow-module channels, query schedules, and an undo mechanism. The language rewrite harmed aggregate success across both suites, whereas the revert actuator was the only intervention with zero net downside on both; better query placement did not convert into more solved episodes.

5.10 A Closed-Loop Test at the Middle of the Continuum

  • The rewrite deficit deepened from thirteen episodes on LIBERO-Spatial to forty-two on LIBERO-Goal, while the revert arm had zero net downside on both suites.The revert contrast against the adaptive arm was direction-consistent but did not survive Holm correction; the authors do not claim an established success advantage.Source: Table 9 · p. 27, §5.10 · pp. 30–31
  • On task 3, fast-only solved 8/10 episodes while periodic and adaptive rewriting solved 2/10 and 4/10; other tasks showed rewrite gains, so pooled success obscured opposite task-level effects.The aggregate contrast nearly cancelled, motivating the per-task readout.Source: §5.10 · pp. 27–28
  • The adaptive trigger spent 55% of queries on the three lowest-success tasks, versus 38% for periodic scheduling, but the arms were statistically indistinguishable at ten episodes per task.The adaptive allocation followed the threshold rule’s prediction, yet did not yield more solved episodes.Source: §5.10 · p. 28
  • Action ranking scored 56/90 against fast-only’s 52/90, while shuffled candidate order scored 57/90 against the critic’s 56/90; neither comparison established value from slow-module choice.The ranked arm’s 524 of 573 selections concentrated on the third candidate, and randomizing order removed that position bias without changing trigger, sampler, or budget.Source: §5.10 · pp. 28–29
  • The progress screen answered “no progress” on 89% of checks and spent twice the adaptive arm’s query budget, reproducing its outcome rather than screening whether a rewrite would help.It screened the state, not the intervention’s value.Source: §5.10 · p. 29
  • Rewrite success stayed below fast-only at k = 10, 25, and 50, while ranking outcomes stayed within a three-episode band across thresholds despite a 1.9× query-rate range.The tested dose changes did not rescue rewriting or reveal a ranking-trigger effect.Source: §5.10 · pp. 29–30
  • Every rewrite arm had a concentrated downside, including 17 fewer episodes than fast-only for periodic rewriting and 12 fewer for adaptive rewriting; one task at ten episodes cannot confirm a tail claim.The shuffled ranking arm exceeded the pre-registered one-episode-per-task bound by one episode on task 7.Source: §5.10 · pp. 30–31
  • No pairwise contrast among the eleven arms survived Holm correction over twelve comparisons; the smallest adjusted value was 0.12.The extension used ninety paired episodes per arm on nine shared tasks; task 8 was lost for the slow arms after a rollout crash.Source: §5.10 · p. 31

5.11 Remaining Integrations

Several closed-loop integrations remain incomplete: the FiS-VLA success-versus-refresh-period study is blocked by memory, while live RoboICL gate replacement awaits its harness and a functioning slow endpoint.

5.11 Remaining Integrations

  • The FiS-VLA closed-loop success-versus-refresh-period comparison remains blocked because the fully fine-tuned policy needs more memory than four consumer GPUs.Its fidelity-level integration is complete, but that closed-loop attempt is described as uninformative for the mechanism.Source: §5.11 · p. 32
  • Live RoboICL gate replacement still requires the RoboDojo harness and a functioning slow endpoint, although its decision-level evaluation is complete.The completed work includes per-decision leak-free scores, a censoring control, a routing frontier, and a high-cost anchor.Source: §5.11 · p. 32
  • The remaining integration agenda includes validating the online router on the deployed FiS checkpoint, reporting a cross-family four-measure table, and testing mixture routing.The online router’s training labels would come from the measured freshness value function; mixture routing would compare intervention types.Source: §5.11 · p. 32

6 Discussion and Limitations

The discussion treats scheduling evidence as conditional on measurement and competence, and interprets closed-loop outcomes as properties of a module–channel–task setting rather than of a slow module alone. It also bounds the reported evidence by the evaluated systems, task suites, and implementation paths.

What the trace analysis does and does not establish

  • Leak-free early-dispatch statistics forecast failure at AUC 0.62, but episode structure can create staleness correlations, so scheduling–outcome claims require controls.Under teacher forcing, which holds episode structure fixed, the decision-level signal falls from 0.646 to 0.592 under matched-cohort control and below chance in Memory.Source: §6 · pp. 32–33

Synthetic testbed scope

  • The synthetic testbed validates scheduling objectives and protocol with the slow module held fixed, not a particular robot system.The paper frames its broader contribution as protocol-first, with success-rate parity a gate rather than the goal.Source: §6 · p. 33

Mid-continuum boundary case

  • At ρ = 1.53, queries follow the threshold rule’s allocation, but rewrite-channel arms do not beat fast-only because rewrites can corrupt instructions on already-solved tasks.The result concerns the module–channel–cost triple; aggregate success at one operating point cannot distinguish query economics from channel effects.Source: §6 · p. 33

Closed-model dependence

  • Hume has no public code and is discussed, with a value-head ablation on open backbones, rather than reproduced as a baseline.More generally, integration claims are restricted to open paths.Source: §6 · p. 33

Implications for trigger design

  • On embedded-latent systems with ρ ≈1, adaptive gains over a well-tuned fixed ratio are bounded and vanish under homogeneous collapse.The paper argues trigger comparisons should report ρ and leak-free calibration alongside call reductions.Source: §6 · pp. 33–34

Scope of the closed-loop arms

  • The LIBERO channel results are bounded to two suites, two slow endpoints, and one operating point; the stall detector’s 15-step, 2-cm setting was not swept.A stronger fast policy or a third suite could change channel rankings; a systematic detector sweep remains future work.Source: §6 · p. 34
  • The second LIBERO suite replicates the channel ordering, with a deeper rewrite deficit and zero net downside for the revert arm on both suites.The reported channel ordering is limited to two suites, two slow endpoints, and one operating point; a third suite or stronger fast policy could change it.Source: §6 · pp. 34–35

Reproducibility and scope disclosures

  • The reported closed-loop extension uses nine common tasks because two slow-arm rollouts crashed during an advisor restart; the object-suite control saturated at zero successes.The disclosed implementation path also required stand-ins or patches for dependencies, cache indexing, and action/proprioception normalization.Source: §6 · pp. 35–36

A Proofs

The proofs distinguish state-based Value of Intervention routing from fixed-ratio, fixed-budget, and history-only schedules, and derive when periodic schedules provide a renewal baseline. The accompanying evidence relates routing performance to collapse variation, trace-signal limits, and measured staleness effects.

Optimality of VoI routing

  • Querying is optimal when VoI(st) exceeds cslow/wfail, comparing the expected failure-cost reduction with the slow-module cost.The expression absorbs continuation-value terms into Δ through dynamic programming.Source: A Proofs · p. 38

Proposition 1 (fixed ratio)

  • A fixed-ratio schedule is optimal only in the degenerate time-invariant case where VoI equals the query cost at every step.When VoI varies over time, querying only above the threshold lowers cost at equal expected query count.Source: A Proofs · p. 38

Proposition 2 (fixed budget best-of-N)

  • A fixed per-step query budget overspends wherever the fast policy is reliable, unlike threshold routing, which spends nothing there.The fixed-budget rule requires collapse probability to exceed a positive constant at every step for optimality.Source: A Proofs · p. 38

Proposition 3 (history gate)

  • When collapse is conditionally independent of schedule history given state, a history-only gate cannot outperform uniform random querying at the same rate.For non-degenerate VoI, state-based threshold routing has higher expected value because it uses state uncertainty.Source: A Proofs · p. 38

Proposition 4 (cost-symmetric baseline)

  • With state-independent collapse, the renewal baseline minimizes per-step cost at interval k* = 2cslow/(wfailπ), making periodic schedules optimal.The expected cost combines query cost per interval with the expected failure cost from collapse persisting until the next query.Source: A Proofs · p. 38

Adaptive routing result

  • When collapse rate varies, threshold routing strictly improves on the best periodic schedule at equal query rate, with gains growing with temporal dispersion.The noise and corruption sweeps measure this gap as estimator quality is degraded.Source: A Proofs · p. 39

Empirical counterpart

  • First-K schedule statistics forecast failure at AUC 0.62, concentrated in early decisions and fading as the observed prefix grows.The RoboICL traces lack observations, so they cannot test the state-conditional independence assumption.Source: A Proofs · p. 39

Measured frontiers

  • The routing frontier shows threshold routing dominates label-blind periods at matched call reduction; measured staleness cost is nonpositive, while latency falls 2.1× across refresh periods.Figure 7 reports flat action MSE across refresh periods; its shuffled-latent control is +11%.Source: Figure 5 · p. 40, Figure 6 · p. 41, Figure 7 · p. 41
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