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Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective

Jian-Feng Cai, Zhengyi Su, Chao Wang

arXiv:2609.06947v1cs.LG

TL;DR

Prior CFG theory examines guidance from complementary perspectives, while this paper develops a particle-level stagewise geometric account of flow matching and CFG. It establishes stage-dependent attractors and schedule-dependent terminal decay, with exact Euler counterparts.

  • Problem

    Prior CFG theory examines guidance from complementary perspectives, yet the paper identifies a remaining theoretical question about guidance.

    Source: §1 · p. 2

  • Method

    The paper develops a stagewise particle-level analysis of attraction and absorption for flow matching and CFG, including explicit Euler dynamics and general time schedules.

    Source: §6 · p. 22, §8 · p. 30

  • Results

    Unconditional paths progress through mean, convex-hull, and local-cluster geometries; under schedule a(t), final-stage distance decays as O(1-a(t)), and CFG recovers local conditional geometry near isolated clusters.

    Source: §1 · pp. 2–3, §8 · p. 30

  • Takeaways & Limitations

    The results provide a unified particle-level geometric account of flow matching and CFG across continuous and discrete sampling.

    Source: §8 · p. 30

    The theory does not explain stage transitions or eventual cluster selection without assuming capture in advance.

    Source: §8 · p. 30

Abstract

from arXiv · show

Flow matching, together with classifier-free guidance (CFG), is widely used in generative modeling, yet much of the theoretical understanding remains distribution-wise. Since practical sampling follows individual trajectories, distribution-level guarantees alone do not fully capture how trajectories interact with the data geometry or how guidance reshapes it. To overcome this limitation, we establish a unified stagewise geometric theory of attraction and absorption for both continuous dynamics and explicit Euler discretization. Specifically, with $t\in[0,1]$ running from noise to data, we show that unconditional flow trajectories are successively attracted toward a neighborhood of the global mean, the data convex hull, and a neighborhood of a possibly nonconvex local cluster. Across these stages, the corresponding distance satisfies a common contraction estimate, yielding an ${O}(1-t)$ decay of the distance in the final stage. For CFG, the same structure persists with an extrapolated mean, an inflated conditional convex hull, and, near the target cluster, the restored local geometry of conditional flow matching. We further show that a general time schedule $a(t)$ replaces the $O(1-t)$ decay by $O(1-a(t))$. Together, these results provide a unified particle-level geometric account of flow matching and CFG across continuous and discrete sampling.

1 Introduction

Existing theory often studies distributional convergence, while practical generation follows individual trajectories and CFG’s geometric effects across sampling stages remain unclear. This paper develops a stagewise particle-level theory for unconditional flow matching and CFG, including explicit Euler sampling and general time schedules.

1 Introduction

  • Particle-level geometry matters because practical generation follows individual numerical trajectories, while existing theories often characterize distributional convergence.The paper identifies how CFG reshapes individual trajectories across stages as an open question.Source: §1 · p. 2
  • Unconditional trajectories are successively attracted toward a global-mean ball, the data convex hull, and, under suitable local conditions, a possibly nonconvex local cluster.The theory establishes attraction and absorption in the latter two regimes.Source: §1 · p. 2
  • CFG reshapes the stagewise geometry from an extrapolated mean to an inflated conditional convex hull, then restores conditional-flow local geometry.The paper characterizes the guided geometry across sampling stages.Source: §1 · p. 2
  • The attraction–absorption picture has exact grid-level counterparts for explicit Euler sampling in both unconditional flow matching and CFG.These results extend the continuous-time estimates to finite-step trajectories.Source: §1 · p. 3
  • A general schedule preserves the ideal trajectory’s spatial path while changing final-stage decay from O(1-t) to O(1-a(t)).The schedule reparameterizes the ideal continuous trajectory.Source: §1 · p. 3
  • The paper positions its results against prior trajectory analyses by preserving possibly nonconvex clusters and extending the geometry to explicit Euler sampling.It also analyzes the original, unscaled flow-matching trajectory, avoiding the rescaling-induced singularity at t = 0 described for Wan et al. (2025).Source: §1 · p. 3

2 Problem Formulation and Notation

The paper formulates unconditional and conditional flow matching as learned time-dependent vector fields generating ODE trajectories from Gaussian noise. It defines the corresponding interpolants and CFG training setup, alongside the geometric and probabilistic notation used in the analysis.

2.1 Flow Matching

  • Flow matching learns a time-dependent vector field, and its unconditional field generates an ODE trajectory initialized with standard Gaussian noise.The vector field is parameterized by time, state, and a conditioning input or the unconditional marker.Source: §2.1 · p. 4
  • Unconditional training samples an interpolant Xt,∅=(1−t)ξ+tY∅, with Gaussian noise ξ, clean data Y∅, and uniform time t.The unconditional clean-data sample is drawn from p∅.Source: §2.1 · p. 4
  • Conditional flow matching uses a conditional ODE initialized from Gaussian noise and an interpolant formed from ξ and a clean sample Yc drawn from pc.The conditional field is trained using that interpolant.Source: §2.1 · p. 5
  • The analysis develops the unconditional case first and obtains conditional results by restricting the data distribution from p∅ to pc.Source: §2.1 · p. 5

2.2 Classifier-Free Guidance

  • CFG trains conditional and unconditional fields using one network, sampling the corresponding clean-data type with probabilities q and 1−q.At sampling, CFG extrapolates between the learned conditional and unconditional vector fields using guidance scale w>1.Source: §2.2 · p. 5

2.3 Basic notations

  • The notation defines distance to a set, diameter, projection when unique, convex hull, closed neighborhoods, Minkowski sums, and scalar multiples.These set operations provide the geometric language used in the paper.Source: §2.3 · pp. 5–6
  • The unconditional support D is assumed bounded, and each conditional support Dc is a nonempty closed measurable subset with positive unconditional mass, as is its complement.Thus D and Dc are compact; conditional local clusters considered later are bounded.Source: §2.3 · p. 6

3 The Ideal Vector Fields and Corresponding ODEs

The paper analyzes ideal unconditional, conditional, and guided flow ODEs through their posterior-mean vector fields and associated objectives. It also presents a smoothed-distance view that motivates the moving geometries used in the stagewise trajectory analysis.

3.1 The Ideal Flow ODEs

  • The ideal unconditional and conditional fields minimize their respective function-space training objectives, and the pair jointly minimizes the CFG objective.The paper focuses on the ideal ODEs generated by these minimizers rather than their neural-network approximations.Source: §3.1 · p. 7
  • On every compact interval [0,T] with T<1, the ideal unconditional and conditional fields are continuously differentiable and their ODEs have unique classical solutions.The spatial Jacobians are bounded on that interval by the estimate stated in Theorem 3.1.Source: §3.1 · p. 8
  • Regularity at the terminal time t=1 requires additional assumptions and separate analysis.Source: §3.1 · p. 8

3.2 The Smoothed-Distance Objective behind the Flow ODE

  • The smoothed-distance term captures the moving geometry tD and converges to terminal hard squared distance as t approaches 1.The full objective determines the ideal flow direction through a gradient identity and can be viewed as a continuation dynamics for the terminal distance problem.Source: §3.2 · pp. 8–9
  • The smoothed-distance viewpoint treats the ideal ODE as time-dependent gradient dynamics and motivates tracking distance to a moving set tS.The selected set changes by stage: a global mean ball, the data convex hull, then a local-cluster neighborhood.Source: §3.2 · p. 9

4 Stagewise Geometry of Unconditional Flow Matching

Unconditional flow trajectories are described by a sequence of moving geometries: a global-mean ball, the data convex hull, and, under local conditions, a neighborhood of a possibly nonconvex cluster. The convex hull supplies coarse global localization, while the mean ball and cluster neighborhood sharpen the early- and final-stage descriptions.

4.1 Early-stage Attraction by the Global Mean

  • Before tmean, trajectories approach a moving ball around the global mean with distance at most (1-t)||x0||; a sufficiently fine initial Euler grid inherits the result.The ball radius trades off localization against the length of the early-stage interval.Source: §4.1 · p. 11

4.2 Early and Intermediate Attraction and Absorption by the Convex Hull

  • Across the convex-hull stage, distance to t Conv(D) contracts at rate 1-t, and after first entrance trajectories remain inside; the Euler scheme has matching grid-level attraction and absorption.This moving hull provides a coarse, end-to-end localization region.Source: §4.2 · p. 12

4.3 Final-stage Attraction and Absorption by the Local Cluster

  • Under local-cluster regularity and suitable final-stage initialization, trajectories approach tB_rΩ(Ω) at rate 1-t and remain there after entry.The theorem requires assumptions on cluster geometry and local data mass, plus separation from the rest of the data.Source: §4.3 · pp. 13–14

4.4 Conditional Flow Matching by Restriction

  • Conditional flow matching inherits the mean-ball, convex-hull, and local-cluster results by restricting the data distribution and geometry to the conditional support; continuous rates remain O(1-t), with Euler counterparts.The local-mass condition and constants are computed under the conditional distribution.Source: §4.4 · p. 15

5 Analysis of Flow Matching with Classifier-Free Guidance Extrapolation

CFG changes the stagewise target geometry: early attraction is centered at an extrapolated mean, intermediate trajectories use inflated conditional convex-hull neighborhoods, and terminal trajectories recover conditional local-cluster geometry. The analysis establishes attraction and absorption results for these regimes and notes corresponding Euler counterparts.

5.3 Final-stage Attraction and Absorption by the Conditional Local Cluster

  • Near a well-isolated conditional cluster, the prediction gap vanishes toward the terminal time, restoring conditional local-cluster geometry with O(1−t) attraction and absorption after entry.The same final-stage mechanism is reported for unconditional, conditional, and CFG flows.Source: §5 · p. 16, §5.3 · pp. 20–21

5.1 Early-stage Attraction by the Extrapolated Mean

  • The early CFG moving ball is centered at the extrapolated mean, and distance to it contracts at the stated early-stage rate.The center is ¯y + w(¯yc − ¯y); increasing w moves it from the unconditional mean toward and beyond the conditional mean.Source: §5.1 · pp. 16–17

5.2 Early and Intermediate Attraction and Absorption by Inflated Convex Hulls

  • CFG trajectories are attracted to, and after entry absorbed by, an inflated conditional convex-hull neighborhood whose radius is controlled by posterior-mean discrepancy.For a fixed trajectory, delaying the tail start cannot increase the required inflation; after capture and under a separation condition, the radius has an explicit exponential bound.Source: §5.2 · pp. 18–19

5 Analysis of Flow Matching with Classifier-Free Guidance Extrapolation

  • The CFG Euler counterparts are collected in the appendix and use the same mechanism as the unconditional discrete arguments.Source: §5.3 · p. 21

6 Flow Matching with a General Time Schedule

The paper studies flow matching under a general time schedule and shows that the schedule changes the clock along the ideal trajectory rather than its spatial path. A nonlinear example yields quadratic terminal decay in the remaining time.

6 Flow Matching with a General Time Schedule

  • The schedule is incorporated into both interpolation states and regression targets, rather than being applied only during sampling.Source: §6 · p. 22
  • A general schedule reparameterizes the ideal trajectory without creating a new spatial curve; it changes the clock used to travel that curve.The time change is incorporated into both interpolation states and regression targets during training.Source: §6 · p. 22
  • For a(t) = t + t^2 − t^3, the final-stage moving distance decays quadratically in the remaining time 1 − t at the ideal continuous level.Source: §6 · p. 22

7 Experiments

The experiments illustrate stagewise attraction and absorption in synthetic Euler trajectories and CFG sampling, then test general time schedules in fixed-step and adaptive settings. The nonlinear schedule improves reported FID when used in training and lowers measured adaptive-solver costs in the reported setting.

7.1 Convex-hull Attraction and Absorption

  • In the synthetic Euler example, distance to the moving scaled data convex hull decreases to zero at first entry and remains zero afterward.The example starts at t0 = 0.3 on a nonconvex point cloud and uses step size 0.05.Source: Figure 5 · p. 23, §7.1 · p. 23

7.2 Final-stage Local-cluster Attraction and Absorption

  • Five Euler trajectories enter their corresponding scaled nonconvex cluster neighborhoods, where their distances decrease to zero and remain zero after entry.The plotted display radius illustrates the mechanism but is not the analytically constructed theorem radius.Source: Figure 6 · p. 24, §7.2 · p. 24

7.3 Classifier-free Guidance Dynamics

  • Increasing w shifts the extrapolated mean beyond the conditional mean, redirects early trajectories, and is followed by trajectories turning toward the conditional cluster Ω5.The experiments use a uniform empirical unconditional distribution on the cluster union and a conditional distribution on Ω5.Source: §7.3 · pp. 24–25, Figure 7 · p. 25

7.3.1 Early Attraction to the Extrapolated Mean

  • For w = 1.5 and w = 3.0, distance to the CFG extrapolated-mean neighborhood decreases and remains below the linear reference (1−t_i)||x0||.The curves are consistent with the discrete early-stage attraction result.Source: §7.3.1 · p. 25, Figure 8 · p. 25

7.3.2 Prediction-gap decay and final cluster attraction and absorption

  • Near the target cluster, the CFG extrapolation term becomes negligible, and the measured prediction gap fits the predicted asymptotic form for both guidance scales.For each tested guidance scale, distance to the scaled target-cluster neighborhood decreases to zero and stays there after first entry.Source: Figure 7 · p. 25, §7.3.2 · p. 26, Figure 9 · p. 26

7.3.3 Image-generation setting

  • On ImageNet-256, moderate guidance improves class alignment, while very large guidance oversaturates; switch-off after 12 or 16 steps stays close to full CFG, unlike after 6.The early-stage visualization shows guidance amplifying class-specific structure in the attractor and samples from the same initialization.Source: §7.3.3 · pp. 26–27, Figure 10 · p. 27, Table 2 · p. 28

7.4 Flow Matching with a General Time Schedule

  • With a(t) = t + t^2 − t^3, residual smoothing is O((1−t)^2), and Scheme C reaches recognizable low-smoothing states at smaller t.Across MNIST, CIFAR-10, and ImageNet 32 × 32, Scheme C has the lowest reported FID, while sampling-only Scheme B has higher FID than Scheme A in all three settings.Source: §7.4 · p. 28, Figure 12 · p. 29, Table 3 · p. 29
  • FID is 5.74 versus 5.67, mean NFE/image is 156.21 versus 120.84, and mean time/image is 10.67 s versus 8.19 s for standard versus nonlinear schedules under DOPRI5.The nonlinear schedule reduces mean NFE/image and wall-clock time/image by approximately 19%–27% across all six reported classes.Source: §7.4 · p. 29, Table 4 · p. 30

8 Conclusion

The paper concludes that flow matching and CFG trajectories follow a stagewise particle-level geometry, with exact explicit-Euler counterparts and a time schedule that reparameterizes the ideal path without changing its spatial curve. It identifies open work on stage transitions, cluster selection, learned-field errors, and weakening or removing guidance.

8 Conclusion

  • The framework describes unconditional trajectories moving from global-mean geometry to the convex hull and then, under local conditions, to a possibly nonconvex cluster; CFG reshapes early and intermediate geometry before recovering conditional local geometry.The CFG correction becomes exponentially small near a well-isolated target cluster, and explicit Euler has exact discrete counterparts.Source: §8 · p. 30
  • The analysis leaves open how stages transition and the eventual cluster is selected without assuming capture, and how learned-field approximation errors perturb attraction and absorption.A principled account of when CFG can be weakened or removed is also left for future work.Source: §8 · p. 30

Appendix A. Guide to the Appendix

  • Appendix B/C establish field regularity and shared tools, D/E prove unconditional and CFG results, and F treats general schedules; continuous proofs use moving-distance estimates while discrete proofs use Euler-update geometry.The appendix dependency map is read from top to bottom.Source: Appendix A · p. 31, Figure 14 · p. 31

B.1 Proof of Theorem 3.1

This section develops the proof foundations for the stagewise analysis, including regularity and existence results and continuous and Euler-discrete tools for attraction and absorption toward scaled targets. It establishes strict distance decrease toward the mean neighborhood in the continuous early stage and provides an affine convex-hull representation and posterior localization for early Euler iterates.

B.1 Proof of Theorem 3.1

  • On every interval [0,T] with T<1, the unconditional, conditional, and CFG vector fields are continuously differentiable and globally Lipschitz in space, yielding unique C1 ODE solutions.The CFG field is the fixed-weight linear combination of the unconditional and conditional fields.Source: §B.1 · pp. 31–33

B.2 Proofs for Section 3.2

  • Compact support gives uniformly positive mass in every fixed-radius neighborhood and bounds the change in distance when the support set is scaled.The scaling bound is controlled by the maximum norm of points in the compact set.Source: §B.2 · p. 33

B.2.1 Proof of Proposition 3.2(ii)–(iii)

  • The smoothed squared distance differs from the ordinary squared distance by a bound vanishing with the neighborhood radius, and converges to it as t approaches 1.For the convex hull of the scaled support, a point can be chosen whose squared distance matches the smoothed squared distance.Source: §B.2.1 · p. 34

C.1 Continuous-time distance calculus for scaled sets

  • For scaled targets, the continuous-time lemma makes distance non-increasing and guarantees absorption once a trajectory enters the target.The lemma assumes projections onto the scaled target are well-defined and a convex subset contains the relevant target point and velocity direction.Source: §C.1 · pp. 35–36

C.2 Discrete-time attraction and absorption for scaled targets

  • The Euler counterpart gives one-step attraction to scaled targets and preserves membership once a discrete iterate enters the target.Its absorption result also covers the initial step when the starting point is zero.Source: §C.2 · p. 37

D.1 Common posterior-mean facts

  • Posterior means of probability measures supported on D lie in Conv(D), which is compact when D is compact.The proof uses separation for the closed convex hull and compactness in finite dimensions.Source: §D.1 · p. 38

D.2 Continuous early-stage mean attraction

  • An a priori bound on unconditional trajectories is used to control posterior weights in the early mean-stage proof.The proof invokes this bound when establishing that the posterior mean remains in a fixed ball around the global mean.Source: §D.2 · p. 38, §D.2.1 · p. 40

D.2.1 Proof of Theorem 4.1

  • The early-stage proof establishes non-entry into the chosen mean neighborhood and shows that the posterior mean stays inside a fixed ball around the global mean.The proof uses that both the posterior mean and global mean belong to Conv(D).Source: §D.2.1 · p. 39
  • The continuous early-stage distance to the scaled mean neighborhood is positive and strictly decreasing through tmean.The attraction argument applies a projection-based contraction estimate to the mean neighborhood.Source: §D.2.1 · pp. 40–41

D.3 Discrete early-stage mean attraction

  • Each unconditional Euler iterate has an affine representation using a point in Conv(D), and early grid points satisfy posterior localization through tmean.The affine representation is stated for every grid point; the localization result applies when ti ≤ tmean.Source: §D.3 · pp. 41–42

D.3.1 Proof of Theorem 4.2

The proofs establish stagewise attraction and absorption bounds for continuous and Euler flow, using a scaled convex target and, in the final stage, attraction toward an inflated cluster neighborhood with O(1-t) decay. Auxiliary lemmas control projections, local mass, and posterior-mean localization near the target cluster.

D.3.1 Proof of Theorem 4.2

  • During the early stage, Euler iterates satisfy dist(zi, ti Brmean(ȳ)) ≤ (1 − ti)∥x0∥.The proof also notes that ∆t ≤ tmean/2 ensures the early grid segment contains the two nonzero grid points t1 and t2.Source: §D.3.1 · p. 43

D.4.1 Proof of Theorem 4.3

  • For continuous flow, distance to t Conv(D) is non-increasing, and trajectories entering the scaled convex hull remain in it thereafter.The proof extends the attraction estimate to s = 0 and establishes absorption from any tconv ∈ (0,1), with the tconv = 0 case handled separately.Source: §D.4.1 · p. 44

D.4.2 Proof of Theorem 4.4

  • For Euler flow, distance to ti Conv(D) is non-increasing, is bounded from the initial point, and absorption persists after any grid point enters the scaled convex hull.The proof establishes a pairwise estimate for 0 ≤ j ≤ i ≤ N with tj < 1, then proves absorption by induction.Source: §D.4.2 · pp. 44–45

D.5.1 Local geometric and posterior-localization lemmas

  • The auxiliary lemmas establish prox-regular projection estimates, a polynomial local-mass radius, an inflated-neighborhood projection formula, and posterior-mean localization near the cluster.The localization bound applies for t ≥ tloc when x lies within distance tR of tΩ.Source: §D.5.1 · pp. 45–48

D.5.2 Proof of Theorem 4.5

  • For t ≥ tloc, distance to the inflated cluster neighborhood is non-increasing and decays as O(1-t).The proof uses the capture condition to extend the estimate throughout the final stage.Source: §D.5.2 · p. 52

D.5.3 Proof of Theorem 4.6

  • Euler attraction makes the scaled-target distance non-increasing and bounds it by O(1 − t_i).The bound’s factor is independent of the update index.Source: §D.5.3 · p. 53
  • Once an Euler iterate enters t_k B_{r_Ω}(Ω), every subsequent iterate remains in its corresponding scaled target set.The proof establishes this by induction using the projection and posterior-localization lemmas.Source: §D.5.3 · p. 53

Appendix E. Proofs for Section 5

  • CFG proofs verify that the extrapolated posterior mean and normalized projection lie in a convex subset of the static target, enabling continuous and Euler attraction and absorption results.The appendix also states an a priori CFG ODE norm bound for times t < 1 when the solution is defined.Source: Appendix E · p. 54

E.1 Proof of Theorem 5.1

The proof establishes a CFG mean-neighborhood bound and then applies a common distance contraction argument. It also derives the CFG non-entry property using an extrapolated mean and a modified diameter bound.

E.1 Proof of Theorem 5.1

  • The CFG non-entry argument follows the analogous calculation with the extrapolated mean and the modified diameter term (w−1)D∅ + wDc.The proof states that this establishes part (i) of the theorem.Source: §E.1 · p. 54
  • The CFG posterior mean lies within the defined radius of the extrapolated mean for t in [0, t̃_mean].At t = 0, the CFG posterior mean equals the extrapolated mean.Source: §E.1 · p. 55
  • On the interval through t̃_mean, the distance to the time-scaled CFG mean neighborhood decreases strictly as time advances.The proof uses the factor (1-t)/(1-s) < 1 for s < t.Source: §E.1 · pp. 55–56

E.2 Proof of Theorem 5.2

The proof uses the CFG extrapolation to establish attraction toward an inflated conditional convex hull and absorption after the trajectory enters its time-scaled neighborhood.

E.2 Proof of Theorem 5.2

  • The conditional and unconditional posterior means lie in their respective data convex hulls, whose diameter bounds their separation by D∅.CFG extrapolates from the conditional posterior mean using their difference, scaled by w−1.Source: §E.2 · p. 56
  • From t_infl onward, the distance to the time-scaled inflated conditional convex hull is non-increasing.The proof applies the projection estimate on the interval beginning at t_infl, including time zero when t_infl = 0.Source: §E.2 · p. 56
  • Once the trajectory enters the time-scaled inflated conditional convex hull, the proof establishes absorption for all later times before 1.The t̃_conv = 0 case is handled by taking s = 0 in the attraction estimate.Source: §E.2 · p. 57

E.3 Proof of Corollary 5.1

The corollary refines CFG attraction to a less inflated conditional convex hull and relates the absorbed trajectory to the conditional data cluster. Its contraction bound decays with the remaining time to 1.

E.3 Proof of Corollary 5.1

  • After absorption, xt/t is within ε_infl(t_infl) + Dc of every y in Dc, yielding a time-scaled bound on distance to the conditional cluster.The proof obtains this from membership of xt in the time-scaled inflated conditional convex hull.Source: §E.3 · p. 57
  • Points outside Dc remain separated from xt/t by at least ρ_infl, producing a corresponding time-scaled lower bound.The proof combines this separation with the conditional-cluster estimate.Source: §E.3 · pp. 57–58
  • The inflation radius ε_infl(t) tends to zero as t approaches 1, allowing a reference time with ε_ref < ε_infl(t_infl).The proof chooses t_ref sufficiently close to 1 to obtain this strict inequality.Source: §E.3 · p. 58
  • For t in [t_ref, 1), distance to the time-scaled B_εref(Conv(Dc)) contracts by the factor (1−t)/(1−t_ref).This is the stated refined-distance estimate for attraction toward the less inflated conditional convex hull.Source: §E.3 · p. 58

E.4 Proof of Theorem 5.3

This part proves stagewise attraction and absorption for conditional CFG, covering final-stage localization and discrete Euler dynamics toward extrapolated means and inflated conditional target sets. It also establishes contraction and preservation of entry into those sets, and bounds the conditional–unconditional posterior-mean gap as their vector fields converge near the endpoint.

E.4 Proof of Theorem 5.3

  • The proof bounds the conditional–unconditional posterior-mean gap and shows both that gap and the corresponding vector-field difference converge to zero as t approaches 1.The vector-field difference is expressed as the posterior-mean gap scaled by 1/(1 − t).Source: §E.4 · p. 60

E.5 Final-stage attraction and absorption in CFG

  • Under the CFG local-cluster regularity assumptions, the final-stage results establish a conditional local-mass radius, conditional localization, and CFG localization near the target cluster.These results apply for sufficiently late times when the trajectory is within the specified neighborhood of the scaled cluster.Source: §E.5 · pp. 60–62, §E.4 · p. 60

E.5.1 Proof of Theorem 5.4

  • Theorem 5.4’s proof shows final-stage distance to the scaled cluster neighborhood is non-increasing and decays as O(1 − t), with absorption once the trajectory enters the neighborhood.The attraction bound holds throughout the final-stage interval; the proof rules out exit from the specified neighborhood.Source: §E.5.1 · pp. 63–64

E.6 CFG Euler stagewise counterparts

  • In the early stage, for ti ≤ t~mean, the CFG Euler iterates satisfy dist(zi, ti B~rmean(ȳcfg)) ≤ (1 − ti)||x0||, bounding distance to the scaled ball around the extrapolated mean.Source: §E.6 · p. 64
  • For the earlier CFG Euler stage, refined distance to the inflated conditional convex hull is non-increasing, and entry into that set guarantees absorption thereafter.The result applies from an index ℓ satisfying the stated stage condition; if z_{k′} lies in the inflated convex-hull neighborhood, all subsequent iterates remain there.Source: §E.6 · pp. 68–69
  • The shifted CFG Euler trajectory’s distance to the inflated conditional cluster neighborhood is non-increasing and bounded by a factor proportional to 1−t_i.The theorem considers a shifted grid initialized at the local start time; the bound is dist(z_i, t_i B_{r̃_{Ωc}}(Ω_c)) ≤ ((1−t_i)/(1−t̃_loc)) dist(z_0, t̃_loc B_{r̃_{Ωc}}(Ω_c)) = O(1−t_i).Source: §E.6 · p. 70
  • Once a shifted CFG Euler iterate enters the inflated conditional cluster neighborhood, every later iterate remains inside it.The absorption guarantee applies when z_k ∈ t_k B_{r̃_{Ωc}}(Ω_c).Source: §E.6 · pp. 70–71

F.1 Proof of Proposition 6.1

The proof establishes that a general time schedule preserves the standard-clock trajectories and transfers the attraction and absorption conclusions to the new clock.

F.1 Proof of Proposition 6.1

  • The ideal-field identity holds for both unconditional and conditional branches under the general schedule.The proof establishes the branch identities and applies the same uniqueness and extension argument to the conditional branch.Source: §F.1 · pp. 72–73
  • On each interval [0,T] with T < 1, the branch fields and their CFG combination are continuous and globally Lipschitz, giving unique ODE solutions.The proof uses the schedule’s monotonicity and a(T) < 1 to establish the bounds.Source: §F.1 · p. 73
  • For unconditional flow, conditional flow, and CFG, uniqueness makes the general-schedule trajectories coincide with the standard-clock trajectories on [0,1).The schedule changes the field by the chain-rule factor while retaining the same initial condition.Source: §F.1 · p. 73
  • The standard-clock attraction estimate transfers to the general schedule, preserving monotonicity and an O(1-a(t)) distance bound.The proof also transfers absorption, applying the substitution to the conditional result and the CFG theorem.Source: §F.1 · p. 73
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