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Velocity Ambiguity Profiles: Time-Resolved Bayes-Risk Diagnostics for Flow Matching

Yunpeng Mei, Xiaowen Zhu, Chenyu Wang, Chenbo Xin, Hongjie Cao, Jiamin Wang, Jie Chen, Gang Wang

arXiv:2610.30107v1cs.LG

TL;DR

Flow matching’s path-averaged loss obscures which interpolation times are limited by intrinsic velocity ambiguity and which remain reducible estimation problems. The paper introduces a time-local Bayes-risk profile and decomposes loss into that floor and reducible excess; guided time sampling lowers velocity loss and global feature FD on CIFAR-100 DINOv2 feature flow, while conditional KID shows no gain.

  • Problem

    Flow matching usually reports loss averaged over interpolation time, obscuring which regions are limited by intrinsic ambiguity and which remain data-limited.

    Source: §1 · p. 1

  • Method

    The paper defines VAP as a time-local Bayes-risk floor and decomposes time-resolved loss into that floor plus reducible excess risk.

    Source: §1 · pp. 1–2

  • Results

    6.78% lower uniformly evaluated velocity loss and 23.6% lower global feature FD accompany guided time sampling on CIFAR-100 DINOv2 feature flow, while macro conditional KID shows no gain.

    Source: §1 · p. 3

  • Takeaways & Limitations

    VAP makes the aggregate flow-matching objective a time-resolved diagnostic for locating ambiguity-limited and data-limited regions.

    Source: §7 · p. 10

    The natural-data result is limited to frozen DINOv2 feature space and makes no pixel-quality, universal-sampler, or uniform-metric-improvement claims.

    Source: §7 · p. 10

Abstract

from arXiv · show

Flow matching trains a continuum of velocity regression problems but usually reports a loss averaged over interpolation time, obscuring which regions are limited by intrinsic ambiguity and which remain data-limited. We introduce the Velocity Ambiguity Profile (VAP), $\mathcal{A}_v(t)$, the time-local Bayes-risk floor of velocity prediction, and decompose time-resolved loss as $L_N(t)=\mathcal{A}_v(t)+R(t;N)$, where only the excess term $R(t;N)$ is reducible by more data or a different estimator. For $N$ training samples and a $K$-component mixture in the structured example, we prove two statistical benchmarks. In an intrinsic local-regression benchmark, $\mathcal{A}_v(t)$ enters the time-dependent noise factor of the standard nonparametric rate. In separated isotropic Gaussian mixtures, the Bayes velocity becomes locally affine, and a label-oracle residual-regression benchmark has parametric excess-risk scale $K\mathcal{A}_v(t)/N$, matched within the same separated residual Gaussian-location subproblem. Controlled Gaussian-mixture experiments validate this diagnostic through a dense transition sweep, calibrated oracle-to-blind and empirical-proxy audits, and a local controlled time-sampler stress test in which a frozen reducible-error proxy reduces reducible loss while cross-cell sensitivity delimits aggressive sampling. On CIFAR-100 DINOv2 feature flow, guided time sampling reduces uniformly evaluated velocity loss by $6.78\%$ and global feature Fr\'echet distance (FD) by $23.6\%$, while class-conditional kernel inception distance (KID) shows no improvement. VAP turns an averaged flow-matching objective into a time-resolved diagnostic for locating ambiguity-limited and data-limited regions.

1 Introduction

Flow matching’s path-averaged loss obscures which times are limited by reducible estimation error and which by intrinsic velocity ambiguity. The paper introduces VAP as a time-local Bayes-risk diagnostic, develops scoped statistical benchmarks, and reports both controlled validation and natural-feature results.

1 Introduction

  • VAP separates the time-local Bayes-risk floor from reducible excess risk, so only the excess can be lowered with more data or a different estimator.The paper frames this as a path-specific diagnostic for identifying ambiguity-limited and data-limited times.Source: §1 · pp. 1–2
  • VAP complements time-aggregated generative-model baselines by asking which interpolation times are data-limited and which are ambiguity-limited.Source: §1 · p. 2
  • The intrinsic local-regression benchmark places VAP in the time-dependent noise factor, while the separated Gaussian-mixture label-oracle benchmark has excess-risk scale KAv(t)/N.In the separated mixture regime, the Bayes velocity is locally affine and the residual scale is matched within a separated residual Gaussian-location subproblem.Source: §1 · p. 2
  • 6.78% lower uniformly evaluated velocity loss and 23.6% lower global feature FD accompany guided time sampling on CIFAR-100 DINOv2 feature flow; macro conditional KID shows no gain.Recall and effective covariance rank also decrease.Source: §1 · p. 3

2 Preliminaries

The preliminaries describe conditional flow matching as regression onto a velocity target along a linear interpolation between data and Gaussian noise.

2 Preliminaries

  • The path interpolates data a and isotropic Gaussian noise ε as xt = tε + (1−t)a for t in [0, 1], with conditional velocity ut = ε − a.Source: §2.1 · p. 3
  • A context-conditioned network vθ(xt, t, c) is trained to regress onto ut using the flow-matching loss.Source: §2.1 · p. 3
  • At inference, actions are generated by integrating the velocity field from Gaussian noise to data with an ODE solver.Source: §2.1 · p. 3

3 Velocity Ambiguity Profile

The section defines VAP as the time-local Bayes risk of velocity prediction, then uses a Gaussian-mixture example to explain its time dependence and separates irreducible risk from estimator excess. It also describes a reference-relative diagnostic for practical use.

3.1 Definition and Interpretation

  • VAP is the conditional velocity variance averaged over the interpolant distribution, equivalently the smallest expected squared velocity error at a fixed time.For t > 0, the action-posterior variance is rescaled by 1/t².Source: §3.1 · p. 3

3.2 Analytic Example: Gaussian Mixture

  • For an isotropic Gaussian mixture, VAP is non-monotone: component ambiguity grows when multiple means remain plausible, while the large-time limit approaches the data covariance scale.At small times the within-component term dominates; intermediate times mix posterior responsibilities.Source: §3.2 · p. 4, §3.1 · p. 4

3.3 Bias-Variance Decomposition of the Time-Resolved Loss

  • The time-resolved test loss decomposes into VAP plus nonnegative excess risk relative to the Bayes-optimal velocity.The decomposition follows from conditional-mean orthogonality.Source: §3.3 · p. 4

3.4 Practical Estimation of Av(t)

  • A largest-budget validation loss is an upper-biased reference for VAP, and the current-model gap to it measures excess-risk reduction relative to that reference rather than the absolute Bayes floor.Consistency toward the floor additionally requires vanishing training excess and a growing validation sample; normalized excess, not raw loss alone, is the clean collapse diagnostic.Source: §3.4 · pp. 4–5

4 Main Theoretical Results

The theory gives two scoped benchmarks for how VAP relates to reducible estimation error, plus a regime-wise normalization principle. The Gaussian-mixture result is specifically a separated label-oracle residual-regression benchmark, not a full-information lower bound.

4 Main Theoretical Results

  • VAP enters the intrinsic local-regression benchmark’s nonparametric rate as a time-dependent noise factor, while the separated Gaussian-mixture residual benchmark has parametric scale KAv(t)/N.The Gaussian-mixture benchmark supplies true component labels and is not a lower bound over all action-aware estimators.Source: §4 · p. 5

4.1 Time-Resolved Nonparametric Benchmark Bound

  • The intrinsic compact local-regression bound recovers the usual exponent only in the fixed-positive-noise, coverage-sufficient regime; its finite-sample coverage term matters at low noise.It is not a minimax claim for the full ambient flow-matching observation model.Source: §4.1 · p. 5

4.2 Interpretive Extension: Adaptive Rates for Structured Data

  • The adaptive-rate and merge-time discussion is interpretive: in separated Gaussian mixtures, posterior mass concentrates on one mode below the stated threshold, while general multimodal merge time is only a heuristic proxy.Source: §4.2 · p. 6

4.2.1 Oracle Result for Gaussian Mixtures

  • Within the separated label-oracle residual-regression subproblem, upper and lower bounds identify KAv(t)/N up to exponentially small separation and count corrections.The result relies on separated mixture cells and an estimator given true component labels and common variance.Source: §4.2.1 · p. 6

4.3 Curve Collapse: A Normalization Principle

  • When excess risk factorizes with a time-independent sample-size exponent, dividing it by VAP removes the leading time dependence, leaving bounded-factor or exponentially small residual variation in the stated regimes.The collapse is exact only for the separated label-oracle residual benchmark; the minimax benchmark supports bounded-factor common-exponent behavior.Source: §4.3 · pp. 6–7

5 Experiments

The experiments test VAP through controlled Gaussian-mixture transitions and audits, a time-sampler stress test, and CIFAR-100 feature flow. Guided sampling improves training and some deployment metrics, with sensitivity and diversity limits.

5 Experiments

  • The experiment program moves from a controlled transition to oracle and proxy audits, then tests proxy-guided sampling and a natural-image feature-flow application.The controlled study uses analytic VAP and canonical reliability filtering; the natural-data application uses frozen DINOv2 features.Source: §5 · p. 7, §5.1 · p. 7

5.1 Canonical Controlled Transition

  • Across five sampled plateau rows at Δ/σ=4–6, the pooled slope stays within roughly 4% of −1; at Δ/σ=15 it attenuates to −0.605.At low overlap, Δ/σ=1.5, slope magnitude shifts from about 0.98 to 0.46 across readout filters, so the result is reported as threshold-sensitive.Source: §5.1 · p. 7, Figure 2 · p. 8

5.2 Oracle-to-Blind and Proxy Calibration

  • The five-row oracle-to-blind audit finds the label-cost identity holds within 0.07%, while blind MLP residuals are 0.64%–1.71%; proxy rank agreement is 0.959–1 under its guard.The proxy is calibrated for cells passing rF < 2, not presented as a universal replacement for the analytic floor.Source: Table 1 · p. 7, §5.2 · p. 8

5.3 Controlled Time-Sampler Stress Test

  • All tested gap-guided arms improve paired reducible loss, with plateau gains of 9.32%–20.21%; aggressive sampling weakens or reverses across cells.Pure α=2 is lowest at the plateau and α=4 near the peak-learning step; α=8 reverses in the late plateau of the largest sensitivity-check cell.Source: §5.3 · pp. 8–9, Figure 3 · p. 9

5.4 Natural-Data Application: CIFAR-100 DINOv2 Feature-Flow

  • 6.78% lower uniformly evaluated velocity loss and 23.6% lower global feature FD are reported for guided sampling on CIFAR-100 DINOv2 feature flow.Macro conditional KID shows no improvement, and lower recall and effective covariance rank indicate a diversity cost; the results are feature-space metrics, not image FID.Source: §5.4 · pp. 9–10, Table 2 · p. 10

6 Related Work

The related work situates VAP alongside flow-matching objectives, imitation-learning theory, schedule methods, and statistical estimation theory, emphasizing its time-resolved diagnostic focus.

6 Related Work

  • Flow-matching and rectified-flow objectives transport noise to data, while imitation-learning work studies data composition and policy error; schedule methods alter the training-time distribution.VAP instead diagnoses the regression problems induced along interpolation time.Source: §6 · p. 10
  • Diffusion and score theory study minimax rates under manifold and smoothness assumptions, while adaptive regression examines intrinsic or compositional structure.The paper frames its intrinsic result as a time-resolved velocity analog and its separated Gaussian-mixture result as a scoped label-oracle residual benchmark.Source: §6 · p. 10

7 Conclusion

The conclusion presents VAP as a time-resolved separation of Bayes-risk floor and reducible excess, supported by scoped theory and controlled experiments. Its natural-data evidence is limited to frozen feature space, and turning diagnostics into a budget rule or intervention policy remains open.

7 Conclusion

  • The paper concludes that VAP separates ambiguity-limited loss from reducible excess, with two scoped benchmarks and controlled and CIFAR-100 feature-flow evidence.The conclusion reports 6.78% lower velocity loss and 23.6% lower global feature FD, but no macro conditional KID gain and a diversity cost.Source: §7 · p. 10

Appendix

  • The appendix maps proofs for the intrinsic and separated-mixture benchmarks, curve collapse, experimental protocols, auxiliary audits, and BEHAVIOR-1K application context.It also lists the assumptions and proofs underlying the decomposition and VAP boundary behavior.Source: Appendix · p. 15, A Basic VAP Facts and Intrinsic… · p. 16

A Basic VAP Facts and Intrinsic Benchmark Proofs

  • The intrinsic benchmark uses compact local regression with controlled design, smoothness, boundary, and noise conditions; its risk characterization is bounded-factor and not a full ambient minimax result.The proof uses an Assouad lower bound and kernel-regression upper bound on compact time intervals where VAP is bounded above and below.Source: A Basic VAP Facts and Intrinsic… · p. 16, §A.1 · pp. 16–17

A.1 Assumptions for Theorem 4

  • Theorem 4 assumes compact manifold support and finite smoothness; its stated regularity conditions are benchmark assumptions, not consequences of Gaussian smoothing.The appendix distinguishes these conditions from the separated Gaussian-mixture theorem's finite-dimensional residual result.Source: §A.1 · pp. 16–17

A.4 Verifiable sufficient conditions for Assumption A5

  • Uniform comparability of pointwise noise to average VAP is structural; it holds under specified covariance bounds but may fail in bridge-like multimodal geometries.The sufficient conditions include strongly log-concave unimodal families and finite mixtures away from their merge region.Source: §A.1 · pp. 16–17, §A.4 · p. 17

A.2 Proof of Proposition 3

  • The decomposition proof uses conditional-mean orthogonality, while the VAP boundary discussion gives finite behavior near t=0 and a non-monotone single-Gaussian profile.At t=1, the interpolant is independent of the data action and VAP equals the trace of its covariance.Source: §A.2 · p. 17, §A.3 · p. 17

A.5 Intrinsic vs. ambient dimension

  • Although interpolants live in ambient space, the intrinsic benchmark rate depends on manifold dimension because the velocity field varies primarily along tangent directions.The appendix states that both the lower and upper bounds operate in the d-dimensional local benchmark; the ambient reduction remains unresolved.Source: §A.5 · pp. 17–18, §A.6 · p. 18

A.6 Problem Setup

  • The problem setup treats fixed-time velocity prediction as heteroscedastic regression with average noise VAP(t); relating its benchmark minimax risk to the full ambient model requires additional projection arguments.That constructive reduction is left to future work.Source: §A.6 · p. 18

A.7 Lower Bound via Assouad’s Method

The appendix establishes scoped lower and upper benchmarks for time-local velocity regression, with the ambiguity profile acting as the regression noise scale. In separated Gaussian mixtures, it analyzes a label-oracle residual-regression benchmark and distinguishes it from the stronger full-supervision information set.

A.7 Lower Bound via Assouad’s Method

  • The Assouad lower bound is for an induced local d-dimensional regression experiment in the compact benchmark, not the full ambient flow-matching observation model.Its Gaussian-noise subclass restriction is needed for the lower bound; the upper bound does not require Gaussian noise.Source: §A.7 · pp. 18–19

A.8 Upper Bound via Stabilized Local Polynomial Regression

  • The stabilized local-polynomial upper bound assumes a compact chart, bounded design density, a uniform population Gram lower bound, and the stated smoothness condition.Its bound includes a random-design coverage-failure term that cannot be dropped in the zero-noise limit.Source: §A.8 · p. 20

A.9 Combining the Bounds

  • On compact time intervals with bounded positive Av(t) and B(t), the lower and stabilized upper bounds share the classical exponent only when the uniform coverage threshold is met.At finite N or as Av(t) approaches zero, the coverage term remains essential, so the result does not claim a global two-sided Av(t)-power law or exact linear factorization.Source: §A.9 · p. 20

B Proof of the Gaussian-Mixture Benchmark

  • In separated Gaussian mixtures, posterior concentration makes the Bayes velocity locally affine up to exponentially small error, yielding a parametric label-oracle residual-regression upper bound and a matching lower bound for that subproblem.The result is not a lower bound over all estimators of the full supervised tuple.Source: B Proof of the Gaussian-Mixture… · p. 21, §B.2 · p. 22

B.1 Setup and Notation

  • The Gaussian-mixture setup uses a K-component isotropic mixture along a linear interpolation path, with target velocity ε − a and Bayes velocity E[ut | xt, t].The time-local Bayes risk is Av(t) = E_xt[tr Cov[ut | xt, t]].Source: §B.1 · p. 21

B.2 Information Sets and Label-and-Action Oracles

  • At N=2000, the residual estimator’s NRres(t; N)/(KAv,within(t)) is within 5.9% of unity at d=16, while the label-and-action to residual risk ratio tracks t2/σ2 t within 8.5%.At t=0.05, the label-and-action oracle is approximately 90× tighter; by t=0.80, the two are within 1.5%.Source: §B.2 · p. 22

B.3 Main Result

This section decomposes velocity ambiguity into within- and between-component contributions and analyzes the transition to the separated regime, where the Bayes velocity is locally affine. In that regime, a label-oracle residual estimator has parametric excess-risk scale K Av(t)/N, matched by a lower bound within the separated residual Gaussian-location subproblem.

B.3 Main Result

  • In the separated regime, a label-oracle residual estimator has excess-risk scale K Av(t)/N, matched within the residual Gaussian-location subproblem up to separation and count corrections.The result applies below the merge-time boundary; it does not prove a sharp rate for blind estimators on general distributions or prescribe practical time reweighting.Source: §B.3 · pp. 22–24

B.5 Discussion: Beyond Gaussian Mixtures

  • For structured distributions, a conjecture proposes time-varying effective smoothness and dimension, with parametric behavior before a transition and nonparametric rates after it.The proposed adaptive-rate extension is interpretive rather than proved; the Gaussian-mixture theorems stand independently.Source: §B.5 · p. 24, §B.6 · p. 25

B.6 Effective parameters and adaptive-rate conjecture

  • The effective parameters seff and deff are phenomenological: observed α(t) does not identify them without additional structural assumptions.For Gaussian-mixture residual regression, seff →∞ and deff = d encode α = 1 as notation, not as an infinite-smoothness limit of Theorem 4.Source: §B.6 · p. 25, §B.7 · p. 25

B.8 Controlled adaptive-rate evidence: trained network and copying estimator

  • In controlled evidence, the trained network has αdata ≈0.94 and αnoise ≈0.5, while the copying estimator tracks the dimension-dependent exponent 2/D.The network’s ordering held across two time samplers and two budget conventions after a minimum-gradient-step floor; this supports, but does not prove, the conjecture.Source: §B.8 · p. 26

B.9 Comparison with Theorem 4

  • The intrinsic benchmark gives exponent 2s/(2s + d) with time dependence controlled by Av(t), while the separated-mixture result exploits finite-dimensional structure for a parametric rate.The two rates address different settings: a worst-case intrinsic nonparametric benchmark versus the separated Gaussian-mixture residual problem.Source: §B.7 · p. 25, §B.9 · p. 26

B.10.1 Posterior Computation

  • The appendix derives a Gaussian-mixture posterior for a given xt and time t, with posterior component responsibilities and component-wise posterior means.These quantities form the posterior used to compute the Bayes velocity and conditional ambiguity.Source: §B.10.1 · p. 27

B.10.2 Velocity Field Closed Form

  • The Bayes velocity is locally affine when posterior responsibilities are effectively dominated by one component, making the posterior-weighted component mean constant in xt.The conditional covariance of velocity equals 1/t^2 times the conditional covariance of a; that covariance decomposes into within- and between-component terms.Source: §B.10.2 · p. 28, §B.10.3 · p. 28

B.10.3 Av(t) Decomposition

  • The velocity ambiguity profile separates into within-component uncertainty and a nonnegative between-component contribution that grows important as modes overlap.For well-separated modes, the between-component term is approximately zero; when effective separation is at most about one, it becomes dominant.Source: §B.10.3 · p. 29

B.10.4 Parametric Regime Upper Bound

  • In the separated regime, label-oracle residual regression has excess-risk scale K Av(t)/N, with exponentially small separation and component-count corrections.The upper bound uses true component labels and OLS within each component; the matching lower bound applies only to the separated residual Gaussian-location subproblem, not the full flow-matching problem.Source: §B.10.4 · pp. 31–32

B.11.1 Merge Time: Rigorous Definition

  • At the merge time, effective separation crosses a chosen threshold, delimiting when single-component posterior dominance is no longer certified.Effective separation is the Mahalanobis separation between the nearest component means in the time-marginal distribution; the merge-time formula applies when minimum separation exceeds the threshold times the noise scale.Source: §B.11.1 · p. 33

B.11.2 Separated Residual-Subproblem Lower Bound in the Parametric Regime

  • The lower bound establishes the matching parametric scale only for the separated residual Gaussian-location subproblem, not for full flow matching outside that reduction.Within each correctly assigned component cell, subtracting the known affine term yields a Gaussian-location experiment for the velocity intercept.Source: §B.11.2 · p. 34
  • In the separated regime, label-oracle residual Gaussian-location minimax risk has scale K Av(t)/N, with separation and component-count corrections in the upper bound.The upper-bound corrections are exponentially small in separation and include component-count effects.Source: §B.11.2 · p. 36

B.11.3 Transition Analysis

  • For an interior transition, width is O(1/sqrt(log K)) under illustrative τ=C√log K, whereas fixed τ=O(1) can span a visible fraction of [0, 1].If the threshold approaches an endpoint, the local width estimate must be rechecked from the exact derivative.Source: §B.11.3 · pp. 36–37
  • Posterior entropy is low under separation and approaches prior entropy under strong overlap; loss of the low-entropy certificate does not itself establish high entropy.For balanced weights in the strong-overlap regime, entropy is at least c′ log K.Source: §B.11.3 · p. 37

B.11.4 Numerical Verification

The numerical checks illustrate when Gaussian-mixture excess-risk curves show parametric behavior and how normalized-risk collapse depends on the regime and its assumptions. The collapse is exact only under specified factorizations or in the separated-mixture benchmark; transitions make it approximate.

B.11.4 Numerical Verification

  • In the d=2, K=4 corner mixture, the predicted transition lies beyond valid time, while across K the three merge readouts move consistently toward earlier merging.For the corner case, the nonparametric regime would require t ≈1.08; across K=2 to 64, the closed-form merge time falls from 0.781 to 0.580.Source: §B.11.4 · p. 38, Table 3 · p. 39

C Proof of Proposition 6 (Curve Collapse)

  • Regime-wise normalization removes leading time dependence when excess risk factorizes with a time-independent sample-size exponent, leaving bounded or small residual variation.Cross-regime transition-width and high-dimensional sharpening claims are heuristic, not theorem-level results.Source: C Proof of Proposition 6 (Curve… · p. 39

C.1 Case 1: Minimax Regime (Approximate Collapse)

  • For intrinsic local regression, normalized curves have the same exponent only when Av(t) stays bounded above and below; their amplitudes can still differ by a bounded factor.The exponent is 2s/(2s+d), and the bounded-factor result applies on compact interior time intervals.Source: §C.1 · p. 39

C.2 Case 2: Adaptive Rate, Single Regime (Approximate Within Regime)

  • Within one adaptive-rate regime, normalized curves collapse exactly conditional on the factorization assumption; in the separated Gaussian-mixture regime, variation is exponentially small.The parametric-regime claim follows from the label-oracle residual upper bound and local affine approximation.Source: §C.2 · p. 39

C.3 Case 3: Adaptive Rate, Across Regimes (Approximate Collapse)

  • When evaluation spans the transition, the exponent changes from 1 in the parametric regime to 2s_eff(t)/(2s_eff(t)+d_eff(t)) in the nonparametric regime, making collapse approximate.A controlled SGD instance exhibits this shift; the CV analysis is heuristic and depends on the fraction of time points in the transition.Source: §C.3 · p. 40

C.4 Case 4: Threshold-Dependent Sharpening

  • For a Gaussian mixture, transition width is O(1/τ); with illustrative τK = C√log K, the heuristic CV bound is O((log K)^−1/4).Stronger high-dimensional scaling requires an additional SNR-normalization assumption.Source: §C.4 · p. 40

C.5 Case 5: GMM Separated-Regime Result

In separated Gaussian mixtures, the residual-risk factor K/N is independent of time, so normalized curves share exponent 1 up to exponentially small corrections. The appendix also presents an auxiliary data-sufficiency visualization and records the controlled experiment protocols and readouts.

C.5 Case 5: GMM Separated-Regime Result

  • In the separated Gaussian-mixture regime, the K/N factor is time-independent, so normalized residual-risk curves share exponent 1 up to exponentially small corrections.This follows from the label-oracle residual parametric upper bound and exponentially small leakage.Source: §C.5 · pp. 40–41

C.6 Convergence wavefront-auxiliary diagnostic

  • The auxiliary convergence wavefront advances toward later times as N grows, and can be visualized with reducible-risk contours over the time-by-sample-size plane.The paper explicitly does not treat this diagnostic as a main experiment or budget-selection rule.Source: §C.6 · p. 41

C.7 Summary

  • The summary distinguishes bounded-factor minimax collapse, conditional exact single-regime collapse, heuristic across-regime collapse, threshold-dependent sharpening, and the separated-GMM exponent-1 result.Outside controlled Gaussian mixtures, the paper does not predict or claim normalized collapse; its empirical collapse test is limited to the controlled setting.Source: §C.7 · p. 41

D Experimental Protocols and Reproducibility Audits

  • The appendix documents controlled synthetic Gaussian-mixture experiments and retains the BEHAVIOR-1K panel as application-context diagnostics.The controlled experiments use JAX and Optax scripts; the BEHAVIOR-1K panel uses an existing evaluation pipeline.Source: D Experimental Protocols and… · p. 41

D.1 Controlled Gaussian-mixture flow matching

  • Controlled flow-matching examples interpolate sampled actions with Gaussian noise and regress velocity using squared error over a fixed evaluation-time grid.The default estimator is a three-layer MLP, with implementation settings specified for the experiments.Source: §D.1 · p. 42

D.2 Analytic floors, readouts, and statistics

  • The readouts compare analytic-floor excess risk with a large-budget empirical proxy, audit oracle-to-blind gaps, and test frozen-proxy time samplers with paired comparisons.The sampling audit uses mixture and pure proxy-weighted arms, with effective-sample-size and mid-time-mass guards.Source: §D.2 · pp. 42–43

D.3 Full row-level data for the d=4 dense controlled transition

The dense-transition appendix checks whether the parametric plateau survives alternative reliability readouts and threshold choices. A separate normalization audit compares the analytic Bayes-risk floor with alternatives on auxiliary mixture rows.

D.3 Full row-level data for the d=4 dense controlled transition

  • Across the dense d=4 sweep, the plateau conclusion persists across post-hoc reliability filters and all threshold pairs retaining at least 40% of cells.The raw-positive and canonical mid-b plateau slopes remain near the parametric scale; 40/40 qualifying threshold pairs place all five plateau slopes in [−1.10, −0.90].Source: §D.3 · p. 43, Table 7 · p. 44, Table 8 · p. 44

D.4 Normalization sanity audit

  • The normalization audit compares four floor choices across auxiliary d=2 phase rows; the full Av floor is tightest except on the overlap row, where cancellation in R explains the crossing.The large-N reference is a 50000-sample MLP.Source: §D.4 · p. 44, Table 9 · p. 45

D.5 Proxy-floor breakdown at d=2 separated

The proxy-floor audit identifies a specific failure regime at d=2 separated, while the three-family panel shows that mixture geometry shapes residual profiles. On CIFAR-100 DINOv2 feature flow, guided sampling improves training velocity loss and global feature FD, but conditional KID and diversity measures temper the deployment result.

D.5 Proxy-floor breakdown at d=2 separated

  • At d=2 separated, proxy rank correlation spans 0.471–0.868 across five-seed pools and is 0.662 with twenty seeds, with most pools disagreeing on the mid-versus-high region label.The exact-R mass gap between the second and third regions is only 0.16% on each reseed.Source: §D.5 · p. 45, Table 11 · p. 45
  • The d=2 proxy breakdown combines a large-N floor sign flip with a nearly flat reducible-error profile, producing rF = 2.70 above the calibrated boundary of 2.Clamping the proxy floor or trimming the audit window recovers rank agreement; moving to d≥4 removes both factors.Source: §D.5 · pp. 45–46
  • The rF = 2 guard is an empirical separator calibrated on seventeen controlled cells, not a universal threshold.Across the audited cells, the two with rF≥2 have rank correlation at most 0.662, while cells below 2 retain at least 0.959.Source: §D.5 · p. 46, Table 13 · p. 46

D.6 Three-family controlled heterogeneity panel

  • At N=1000, three controlled mixture geometries have distinct residual-profile peaks at t≈0.43, 0.34, and 0.78, with minimum pairwise normalized L2 distance 0.2059.The descriptor is qualitative rather than a hypothesis test.Source: §D.6 · p. 47, Table 14 · p. 47
  • The high-entropy K=8 profile peaks earlier because more components reach near-uniform posterior at smaller t, while the partial-merge profile peaks near the t=0.80 readout boundary.The partial-merge family has between-pair separation Δ/σ=30 and within-pair separation Δ/σ=3.Source: §D.6 · p. 47

D.7 CIFAR-100 DINOv2 feature-flow materials

  • The VAP-guided arm lowers uniformly evaluated velocity loss by 6.78%, while global feature FD falls by approximately 23.6% and macro conditional KID does not improve.The comparison is in feature space, not pixel-space image quality.Source: §D.7 · p. 48, Table 15 · p. 48

D.8 Cross-cell d-axis ladder of the time-sampler stress test

The cross-cell sampler test finds that mixture arms and lower pure exponents retain the reducible-loss reduction, while the most aggressive exponent reverses at the largest mode count. A separate oracle sanity audit confirms that the closed-form label-and-action estimator closely matches the analytic floor and follows its Cramér-Rao rate.

D.8 Cross-cell d-axis ladder of the time-sampler stress test

  • At K=40, d=20, pure α=8 reverses to +11.93% reducible loss in the late plateau, while mixture arms and α∈{1,2} remain negative across all three cells.The α=8 result has sign distribution 3/10 and pperm=0.095; lower exponents have 10/10 signs and pperm<0.005.Source: §D.8 · p. 49, Table 17 · p. 50, Figure 7 · p. 50
  • The sampler’s usable upper exponent appears to shrink with mode count K, although the reported envelope is descriptive rather than a hypothesis test.Pure α=4 remains negative across cells but weakens at K=40 to −11.45%, versus −19.15% in the canonical cell.Source: §D.8 · pp. 49, 51

E.1 Label-and-action Cramér-Rao sanity

  • At N=2000, oracle_ols matches the analytic within-component floor within 0.014%–0.044%, and its empirical excess-to-Cramér-Rao ratio is 0.93–1.15 across five rows.These are implementation-sanity checks, not standalone novel claims.Source: §E.1 · p. 51, Table 18 · p. 51, Table 19 · p. 51

E.2 Blind-Attainability Panel

The blind-attainability audit finds that closed-form blind EM generally tracks the target rate, while the neural blind arm reaches a small residual relative to the within-component floor. A two-stage estimator attains the label-oracle rate only in the separated regime; the BEHAVIOR-1K panel shows lower loss with more demonstrations but task-dependent timing of improvement.

E.2 Blind-Attainability Panel

  • Blind EM matches the target rate band on four of five rows, while the neural blind MLP residual at N=2000 is below 1.8% of within(t) on every row.The partial-merge EM boundary has α=1.36 with bootstrap CI [1.18, 1.67]; the MLP is assessed by floor magnitude rather than a single rate.Source: §E.2 · p. 52, Table 20 · p. 52

E.3 Empirical Sweep of the Two-Stage Estimator

  • For Δmin/σ≥8, the two-stage estimator reaches the label-oracle residual-rate band, but at Δmin/σ≤6 its rate falls to α∈[0.17, 0.34] as misclassification adds an N-independent residual.The separated-to-merge transition is approximately Δmin/σ=6 to 8 in this setup and is not extrapolated beyond the swept range.Source: §E.3 · p. 53, Table 21 · p. 53

F BEHAVIOR-1K Materials

  • The BEHAVIOR-1K appendix records application-context diagnostics and representative simulation-task descriptions, which are not primary evidence for the controlled claims.The diagnostic panel uses π0.5 fine-tuning profiles across fifteen tasks.Source: F BEHAVIOR-1K Materials · p. 53

F.1 BEHAVIOR-1K diagnostic implementation

  • The BEHAVIOR-1K diagnostic implementation evaluates time-resolved velocity MSE on the existing pipeline, with its simulation fine-tuning hyperparameters listed in Table 22.The panel uses a random_seed42 validation split and forbids the corresponding training pools for N=10, 25, and 100.Source: §F.1 · p. 53, Table 22 · p. 54

F.2 Leakage-free diagnostic panel

  • All fifteen leakage-free BEHAVIOR-1K tasks improve in mean loss from N=10 to N=100, but the temporal location varies and profiles fall into three descriptive families.The families are near-uniform improvement, high-time reversal, and smooth no-reversal bottlenecks; they are not benchmark-wide task-family labels.Source: §F.2 · p. 54, Table 23 · p. 54, Figure 8 · p. 55

F.3 Representative task descriptions

The representative BEHAVIOR-1K tasks illustrate varied application contexts, from coordinated bimanual manipulation to long-horizon, multi-stage handling. Their descriptions specify task challenges, randomized initial conditions, and inference limits.

Turning on a radio

  • Turning on a radio requires locomotion and dual-arm coordination, with one gripper stabilizing the device while the other presses its button.The radio pose varies across rollouts, and inference is limited to 6,000 steps.Source: §F.3 · p. 55

Picking up trash

  • Picking up trash pairs long-horizon navigation with asymmetric bimanual roles: one arm carries the container while the other repeatedly collects three cans.Objects and the initial base pose are randomized; the inference budget is 13,000 steps.Source: §F.3 · pp. 55–56

Spraying fruit trees

  • Spraying fruit trees combines mobile manipulation with bimanual tool use, requiring one arm to steady the sprayer while the other operates its switch.The sprayer, trees, and starting base pose vary across rollouts; the horizon is 10,000 steps.Source: §F.3 · p. 56

Cooking hot dogs

  • Cooking hot dogs requires long-horizon planning across a refrigerator, tabletop staging area, and microwave, including buffering one item before transfer.Initial object and base poses vary, with a 15,000-step inference horizon.Source: §F.3 · p. 56

Washing a baseball cap

  • Washing a baseball cap requires navigating with both grippers occupied and placing two caps into a narrow washer drum without collision.Cap and base poses are randomized, with inference limited to 10,000 steps.Source: §F.3 · p. 56

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