Velocity Scaling in Flow Matching

Youssef Saied; François Fleuret

cs.CVOct 7, 2026

Velocity Scaling in Flow Matching

Authors: Youssef Saied, François Fleuret

Organizations: University of Geneva Geneva, Switzerland · University of Geneva and Meta FAIR Geneva, Switzerland

Abstract

Scaling a learned flow-matching velocity field $v_θ$ by a gain $γ(t)$ was recently shown to greatly improve generation quality. Prior work argued that velocity fields trained with mean-squared error (MSE) systematically underestimate velocity magnitude and that scaling corrects this error. We show that MSE training does not create a velocity-magnitude deficit. We find instead that velocity scaling reduces population time lag: sampled states at model time $t$ resemble training states from an earlier time. Velocity scaling and moving model time back are two ways to address this population time lag. Across architectures and model sizes, measuring population time lag and using it to select a gain greatly improves generation quality, reducing FID from 28.0 to 12.2 (estimated by linear interpolation between FID measurements at neighboring gains) on ImageNet-256 at NFE 25 without guidance.

Figures & tables

Figure 1: Trajectories sampled with an unguided stochastic Euler sampler. Within each example, all rows share the same class label, initial latent, and Brownian path. The coarse unscaled path develops the object later in the sequence than the finer path; constant gain 1.15 shifts the coarse NFE-25 trajectory toward the finer timing. Among the tested gains, a sweep with 50,000 images per gain found the lowest FID at gain 1.15 for NFE 25.

Figure 2: Population time lag under constant gains for unguided SiT-XL/2 and ADM U-Net at NFE 25. Positive lag means generated states read as earlier than genuine states; zero is the timing target. Each curve uses 1,280 paired paths, and shading gives pointwise 95% whole-trajectory bootstrap intervals conditional on the chosen gain, trained probe, and genuine baseline. The same paths select the lag-based gain and display its lag. The lowest-FID gains come from separate 50,000-image gain grids.

Figure 3: Gain selection changes with numerical accuracy, guidance, and sampling system. Panel (a) shows that the lowest-FID gain across four ImageNet SiT models moves toward one as NFE rises. Panel (b) shows that stochastic Heun selects a lower gain than Euler at a matched budget of 50 model evaluations. Panel (c) shows that the directly fitted lag-based gain and the lowest-FID gain fall as classifier-free guidance rises, but the lowest-FID gain falls faster. Each guided scalar fit uses 1,280 paired paths at NFE 50. Panel (d) shows that decoder-free CIFAR-10 selects gain one under fixed-step Euler and adaptive RK45. Each underlying FID evaluation uses 50,000 images per gain and the same random inputs across gains within each curve.

Figure 4: Direct scalar lag gains and lowest-FID gains for unguided ImageNet-256 SiT-XL/2 and ADM U-Net. Labels give NFE; horizontal bars are paired 95% intervals selecting the best gain among measured candidates in 2,000 resamples of 1,280 paths, with the trained probe and genuine baseline fixed. Vertical coordinates are minima from existing 50,000-image FID grids, with no continuous FID-gain refinement. Each line tests whether the multiplier fitted at NFE 25 describes the gains at the other NFEs. Distances in gain do not measure FID differences. SiT uses uniform stochastic Euler and ADM uses Euler–Maruyama with a fixed final Mean step.

Model	NFE	Unscaled FID	Lag-gain FID (interpolated)	$\kappa$ -adjusted FID	Lowest measured FID
SiT-S/2	25	71.1	50.9	44.4	44.4
	250	50.6	49.9	44.1	43.8
SiT-B/2	25	47.9	30.2	25.9	25.9
	250	29.1	28.5	24.8	24.8
SiT-L/2	25	30.2	16.1	13.4	13.4
	250	14.6	14.4	12.2	12.2

Table 1: ImageNet-256 FID at NFE 25 and 250; lower is better. Lag-gain FID is linearly interpolated between the measured gains bracketing the unrounded

\gamma_{\rm lag}(N)

. The

\kappa

-adjusted column gives measured FID at the closest tested gain to

\kappa\gamma_{\rm lag}(N)

, including gain one. Each model’s

\kappa

is fitted at NFE 25 and reused at NFE 250. Each measured FID uses 50,000 paired images within its model and NFE; these summaries have no computed uncertainty intervals. SiT uses uniform stochastic Euler and ADM uses Euler–Maruyama with a final Mean step.

Figure 5: Scaling gives larger proportional variance increases to directions with larger real-data variance. We compare gain 1.05 with unscaled sampling for unguided SiT-XL/2 and a matched Gaussian flow under stochastic Euler sampling at NFE 250. Directions are grouped by held-out real-data variance. Each point shows the geometric-mean scaled-to-unscaled variance ratio within a group, divided by that across all directions and expressed as a percentage change from one. Negative values indicate a smaller proportional increase than the overall increase. SiT uses 1,280 paired endpoints per gain and 1,024 real-data latents; Appendix H.2 gives the covariance-slope measurements across NFEs and their uncertainty.

Appendix figures & tables30 assets

Supplementary material from the paper’s appendix.

Appendix

Question	Main finding	Where
Is scaling repairing underestimated velocities?	The population MSE field gives exact transport. The tested pretrained SiT field has an MSE-optimal scalar multiplier near one on genuine training-path states.	§ 4 ; Appendices B , F.1
Why can coarse sampling produce lag?	Coarse integration can miss intermediate refinement. The two-step Euler example shows how missing sample-specific image structure registers as lag.	§ 7.1 ; Appendix D.3
Can we measure this lag?	Comparing mean probe readings on generated and genuine states reveals systematic population time lag, which persists after global scale normalization.	§ 5.1 ; Appendix D
How can scaling help states catch up?	A first-order argument explains how scaling advances lagging states toward their scheduled time; moving model time backward addresses the same mismatch. Controls show that much of the benefit requires the model to predict new velocities at the advanced states.	§§ 5.2 – 5.3 , 6 ; Appendices D.4 – D.5 , G
Can lag select a useful gain?	Minimizing measured lag selects gains without an image-quality metric and improves coarse-sampling FID across the tested ImageNet models. The CIFAR-10 model has little lag and favors gain one.	§ 5.4 ; Appendix E
What determines the useful gain beyond lag correction?	The useful gain depends on solver accuracy, guidance, and the image metric. In unguided ImageNet experiments, one multiplier per model roughly relates lowest-FID gains to lag-based gains across NFEs. For unguided SiT-XL/2, sharpening predicts gains with estimated FID within 0.2 of the best measured value.	§ 7 ; Appendices E.3 , H – I

Appendix

Table 2: Questions, main findings, and supporting sections.

Data use	Representation	Count	Source or reference
ImageNet generation	$3\times 256\times 256$ RGB image decoded from a $4\times 32\times 32$ latent	50,000 generated images	Fixed ADM evaluator references
ImageNet probe	$4\times 32\times 32$ latent	12,800 training; 3,200 calibration; 16,000 held out	Disjoint ImageNet training images
CIFAR-10 generation	Centered $3\times 32\times 32$ RGB pixels	50,000 generated images	Official 1-Rectified Flow reference statistics
CIFAR-10 probe	Centered $3\times 32\times 32$ RGB pixels	16,000 training; 3,200 calibration; 4,000 held out	CIFAR-10 training images

Appendix

Table 3: Data representations, sample counts, and reference data. ImageNet probe counts apply separately to the SiT and ADM U-Net probes. Probability calibration uses genuine states that are separate from probe training and held-out evaluation.

Sampler	Update	Budget	Experiments
Hybrid SDE	SiT SDE; $N-1$ stochastic Euler updates through $t=0.96$ , then one Mean update; the public setting uses $N=50$ ( Ma et al., 2024 )	25–250 calls	Changed-state response and direction controls; model-time correction; guidance
Uniform stochastic Euler	Equation 25 on a uniform grid ( Ma et al., 2024 )	25–250 calls	Population time lag and gain grids; changed-state response across model sizes
Deterministic Euler ODE	Equation 24	50 or 250 calls	ODE response; U-Net and CIFAR-10 transfer
Euler–Maruyama with Mean	LFM SDE with $w_{s}=\sigma_{s}$ and one Mean update ( Dao et al., 2023 )	25–250 calls	ADM U-Net transfer
Velocity-derived SDE	Equation 27	25–250 calls	CIFAR-10 SDE gain curves and lag calibration
Stochastic Heun	Predictor–corrector update defined below	50 calls	Euler–Heun comparison

Appendix

Table 4: Sampler variants used in the experiments. The budget column gives fixed model-call counts; adaptive RK45 chooses its own number of calls.

Probe	Training states	Calibration states	Held-out states	Epochs	Held-out error
SiT, ImageNet	12,800	3,200	16,000	30	0.0258
ADM U-Net, ImageNet	12,800	3,200	16,000	30	0.0250
Rectified Flow, CIFAR-10	16,000	3,200	4,000	20	0.0141

Appendix

Table 5: Probe datasets and held-out accuracy. All states come from genuine linear paths; generated states are never used for training or calibration. Counts give separate training, probability-calibration, and held-out sets. Calibration states are ordinary genuine states used only to select the scalar softmax temperature. Held-out error is the mean absolute difference between posterior-mean probe time and known path time on the unit time scale; lower is better.

Reduction in $t$	Probe change from reducing $t$	Probe change from perpendicular perturbation	Difference [95% interval]
0.025	$-0.0248$	$-0.00130$	$+0.0235\;[+0.0234,+0.0236]$
0.050	$-0.0499$	$-0.00519$	$+0.0447\;[+0.0445,+0.0449]$

Appendix

Table 6: SiT raw-state probe responses on 11,200 held-out genuine ImageNet states. Entries give mean changes in estimated time on the unit time scale, measured as the perturbed estimate minus the original estimate. The difference is the perpendicular response minus the response to reducing

t

. Intervals use 2,000 paired bootstrap resamples within time bins. A response close to

-\Delta t

after reducing

t

, together with a perpendicular response near zero, supports sensitivity to path time.

Method	FID
Unscaled sampling	22.7
Model-time correction	16.7
Velocity scaling, gain 1.075	13.2

Appendix

Table 7: ADM U-Net image quality under the LFM Euler–Maruyama sampler described in Appendix C at NFE 50. Lower FID is better. Each method contains 50,000 ImageNet-256 images with exact class-label, initial-latent, and Brownian pairing. Relative to unscaled sampling, model-time correction changed FID by

-6.0\;[-6.2,-5.9]

; relative to gain 1.075, it changed FID by

+3.5\;[+3.4,+3.7]

. Each contrast is model-time correction minus the other method, and brackets are paired 95% percentile intervals from 2,000 common-row influence resamples with the real reference fixed.

Model	NFE	Lag-based gain [95%]	Lowest-FID gain	Unscaled lag RMS	Scalar lag RMS	Lowest-FID lag RMS
SiT-S/2	25	$1.074\;[1.070,1.077]$	1.150	25.8	6.0	23.9
	50	$1.033\;[1.029,1.034]$	1.100	12.1	3.7	21.9
	100	$1.014\;[1.010,1.017]$	1.075	5.4	2.4	20.4
	250	$1.004\;[1.000,1.007]$	1.050	2.7	2.3	16.1
SiT-B/2	25	$1.073\;[1.070,1.077]$	1.125	24.9	5.4	16.5
	50	$1.032\;[1.029,1.034]$	1.075	11.5	3.3	14.0

Appendix

Table 8: Direct scalar lag calibration across the six models at NFE 25, 50, 100, and 250. SiT and ADM use unguided ImageNet-256 sampling; NCSN++ uses CIFAR-10. Gains are dimensionless; brackets give paired 95% gain intervals from 2,000 whole-trajectory draws over 1,280 paths per candidate, with the trained probe and genuine baseline fixed. The lag columns give RMS population probe-time mismatch multiplied by

10^{3}

; lower is better. Unscaled uses gain one, scalar uses the lag-based gain, and lowest-FID uses the lowest measured FID gain from a 50,000-image grid. These paths were used for gain selection; Table 9 gives separate fresh-path validation. SiT uses uniform stochastic Euler; ADM uses Euler–Maruyama with a final Mean step; CIFAR-10 uses Equation 27 .

NFE	Lowest-FID gain	FID	Next higher gain	FID	$\Delta$ FID [95%]
25	1.125	14.7	1.150	15.0	$+0.3\;[+0.2,+0.4]$
50	1.075	13.2	1.100	13.3	$+0.1\;[+0.1,+0.3]$
100	1.075	12.9	1.100	14.3	$+1.3\;[+1.2,+1.4]$
250	1.050	12.8	1.075	13.5	$+0.7\;[+0.6,+0.8]$

Appendix

Table 11: FID at the lowest-FID gain and next higher tested gain for the unguided ImageNet-256 ADM U-Net under LFM Euler–Maruyama sampling with a deterministic final Mean step; lower FID is better. The lowest-FID gain minimizes measured FID over the grid in Figure 6 . The contrast is

\Delta\mathrm{FID}=\mathrm{FID}_{\rm higher}-\mathrm{FID}_{\rm lowest}

, so positive values favor the lowest-FID gain. Contrasts use unrounded FIDs; brackets give paired 95% intervals from 2,000 common-row influence resamples with the real reference fixed. Each gain uses 50,000 images paired by class label, initial latent, and Brownian path within each NFE.

Model	NFE 25	NFE 50	NFE 100	NFE 250
SiT-S/2	76	55	32	11
SiT-B/2	81	62	36	13
SiT-L/2	84	66	36	9
SiT-XL/2	82	63	42	19
ADM U-Net	84	67	43	24
Mean	81	63	38	15

Appendix

Table 12: Estimated FID recovery (%) at the lag-based gains for unguided ImageNet-256 sampling; larger is better. Recovery is

[\mathrm{FID}(1)-\widehat{\mathrm{FID}}(\gamma_{\rm lag})]/[\mathrm{FID}(1)-\mathrm{FID}(\gamma^{\star}_{\rm FID})]

. Recovery is computed from unrounded FIDs. Table 10 gives the unscaled, interpolated lag-gain, and lowest measured FIDs. Each measured FID uses 50,000 paired images; SiT uses uniform stochastic Euler and ADM uses Euler–Maruyama with a final Mean step. The means weight models equally. No FID or interpolation-error intervals have been computed.

Model	NFE	Predicted gain	Lowest-FID gain	Signed error
SiT-S/2	50	1.107	1.100	$+0.007$
	100	1.086	1.075	$+0.011$
	250	1.076	1.050	$+0.026$
SiT-B/2	50	1.082	1.075	$+0.007$
	100	1.062	1.050	$+0.012$
	250	1.053	1.050	$+0.003$

Appendix

Table 13: Full-gain predictions for the five unguided ImageNet models. Each model’s multiplier is fitted at NFE 25 and reused at the three higher NFEs. The signed error is predicted gain minus lowest-FID grid gain; smaller absolute error is better. Each FID grid point uses 50,000 paired images within its model and NFE. These dimensionless gain-error summaries have no computed uncertainty intervals and do not measure FID at the predicted gains. The SiT-XL/2 FID estimates quoted in the text use linear interpolation at NFE 50 and 100 and the nearest tested gain at NFE 250.

Model	$\kappa$	Full-gain error	Excess error
SiT-S/2	1.071	0.014	0.040
SiT-B/2	1.048	0.007	0.030
SiT-L/2	1.049	0.007	0.032
SiT-XL/2	1.068	0.014	0.037
ADM U-Net	1.044	0.008	0.034

Appendix

Table 14: Gain-rule comparison for the five unguided ImageNet models. Each rule fits one coefficient at NFE 25:

\kappa\gamma_{\rm lag}

1+\alpha(\gamma_{\rm lag}-1)

. The two error columns give the mean of the absolute differences between predicted gain and lowest-FID grid gain at NFE 50, 100, and 250; lower is better. All gains are dimensionless and use the unrounded scalar fits and lowest-FID gains in Table 8 . These summaries have no computed uncertainty intervals.

Paths per candidate	Mean absolute gain change $\times 10^{3}$
128	3.30
256	2.23
512	1.37
1024	0.55

Appendix

Table 15: Sensitivity of scalar gain selection to trajectory count across all six models and four NFEs. For each setting and count, 200 paired subsets of the saved 1,280-path candidate bank are used to refit the lag minimum on interpolated population lag curves. The second column is the mean absolute difference from the 1,280-path fitted gain, multiplied by

10^{3}

, averaged equally over the 24 settings and 200 subsets; lower means greater stability. The trained probe and genuine baseline are fixed. These subsampling summaries have no computed uncertainty intervals and do not include fresh adaptive candidate selection.

Training seed	Gain	Unscaled lag RMS	Scalar lag RMS	Reduction [95%]
2026072307	1.04	13.6	5.0	$+8.6\;[+7.7,+9.4]$
2026072308	1.04	12.9	4.2	$+8.7\;[+7.6,+9.6]$
2026072309	1.04	11.8	4.4	$+7.5\;[+6.4,+8.4]$

Appendix

Table 16: Direct scalar calibration with three probe-training seeds for unguided SiT-XL/2 at NFE 50. Each probe selects a gain at resolution 0.01 from shared candidates evaluated on 256 paired calibration paths. The identity and selected gains are evaluated on 1,280 separate fresh paired paths; the genuine baseline is recomputed for each probe. Lag is RMS population probe-time mismatch multiplied by

10^{3}

, and lower is better. The reduction is unscaled minus scalar lag; brackets give paired 95% whole-trajectory bootstrap intervals from 2,000 draws. Equal selections at 0.01 resolution do not imply zero variation in the continuous optimum. No image metric is used.

NFE	Unscaled	Fitted schedule	Direct scalar
25	0.0254	0.0046	0.0058
50	0.0121	0.0020	0.0039
100	0.0061	0.0026	0.0026
250	0.0037	0.0023	0.0029

Appendix

Table 17: Population time lag RMS for unguided SiT-XL/2 on ImageNet-256 under uniform stochastic Euler sampling. Values are the RMS across saved times of the genuine-minus-generated mean raw-state probe reading, on the unit time scale; lower is better. Each schedule uses 256 calibration trajectories and is evaluated on 1,280 disjoint paths paired across interventions by class label, initial latent, and Brownian path. The direct scalar is selected and evaluated on those same 1,280 paths. The constant gains are the direct scalar lag minima,

\gamma_{\rm lag}

, from Table 8 . This comparison changes both average gain and schedule shape; the matched-average FID comparison above isolates shape. The table reports point estimates.

Figure 7: Held-out scalar multiplier fitted to the pretrained SiT-XL/2 velocity at 20 model times using 16,000 ImageNet image–noise pairs. The top panel expands the range around one; the bottom panel uses the scale that separates the fitted curve from SSC. The curve has mean 0.999, and the shaded region is a simultaneous 95% band.

Dynamics	NFE	Unscaled $E_{N}$ [95%]	SSC $E_{N}$ [95%]
ODE	25	$0.059\ [0.055,0.064]$	$0.071\ [0.068,0.074]$
ODE	250	$0.004\ [0.004,0.005]$	$0.052\ [0.050,0.055]$
SDE	25	$0.096\ [0.090,0.101]$	$0.067\ [0.063,0.072]$
SDE	250	$0.008\ [0.007,0.009]$	$0.065\ [0.063,0.068]$

Appendix

Table 18: Endpoint error under solver refinement for SiT-XL/2 with Euler integration.

E_{N}

is mean squared latent-coordinate endpoint error relative to the fine unscaled reference on 1,024 matched trajectories; lower is closer, and brackets are paired bootstrap 95% intervals.

Dynamics	Gain	Unscaled $e$	Scaled $e$	Change [95%]
ODE	SSC	$3.58\times 10^{-7}$	$1.83\times 10^{-6}$	$+1.48\ [+1.46,+1.49]$
SDE	SSC	$4.10\times 10^{-5}$	$4.45\times 10^{-5}$	$+3.56\ [+3.53,+3.60]$
SDE	Fitted	$4.10\times 10^{-5}$	$4.06\times 10^{-5}$	$-0.33\ [-0.34,-0.32]$

Appendix

Table 19: Isolated one-step numerical error for unguided SiT-XL/2 at NFE 50.

e

is squared latent-coordinate error averaged over all 50 steps and 1,024 matched validation trajectories; lower is closer to the fine unscaled reference. The change is

10^{6}(e_{\rm scaled}-e_{\rm unscaled})

, with negative values indicating lower error. Brackets give paired 95% intervals from 2,000 whole-trajectory bootstrap resamples.

Quantity	Euler	Heun	Contrast	95% interval
Strong endpoint error	0.208	0.148	$-0.060$	$[-0.066,-0.055]$
Population time lag RMS	0.019	0.005	$-0.014$	$[-0.014,-0.013]$

Appendix

Table 20: Matched-budget solver diagnostics on 1,280 exactly paired unguided SDE trajectories. Euler uses 50 steps and stochastic Heun uses 25 predictor–corrector steps, for 50 network evaluations with both solvers. Both endpoints are compared with the same-path 1,000-step stochastic-Heun reference. Lower is better for every row. The contrast is Heun minus Euler; brackets give paired 95% trajectory-bootstrap intervals. Strong endpoint error is root-mean-square error in latent units per coordinate, and population time lag is measured on the unit time scale. Population time lag RMS is the root mean square over saved times of the population-mean probe mismatch.

Endpoint property	Paired set	NFE	Intervention	Unscaled FID	Intervention FID	FID reduction [95%]
Global radius	A	50	Velocity scaling during sampling	13.8	7.6	$+6.2\;[+6.1,+6.4]$
	A	50	Endpoint-only control	13.8	13.8	$-0.0\;[-0.1,-0.0]$
Variance in 256 directions	B	100	Velocity scaling during sampling	11.5	6.8	$+4.7\;[+4.6,+4.9]$
	C	100	Endpoint-only control	11.6	11.6	$+0.0\;[0.0,+0.1]$
Global latent mean and covariance	C	100	Velocity scaling during sampling	11.6	6.8	$+4.8\;[+4.7,+4.9]$
	C	100	Endpoint-only control	11.6	12.3	$-0.7\;[-0.8,-0.7]$

Appendix

Table 21: FID for velocity scaling during sampling and endpoint-only controls on 50,000 ImageNet-256 images per intervention; lower is better. Rows with the same paired-set letter use the same images. Paired set A uses the public hybrid SDE sampler at NFE 50; paired sets B and C use uniform stochastic Euler sampling at NFE 100. The contrast is unscaled FID minus intervention FID, so a positive value favors the intervention; brackets give paired 95% intervals. Paired set A intervals use 2,000 common-row first-order FID influence resamples with the real reference fixed;

-0.0

denotes a negative value smaller than 0.05 in magnitude.

Model	Sampler, NFE, gain	FID: unscaled $\rightarrow$ frozen $\rightarrow$ full	FID-reduction ratio [95%]
SiT-S/2	Uniform stochastic Euler, 25, 1.07	$71.3\rightarrow 66.3\rightarrow 51.3$	24.9% $[24.4,25.4]$
SiT-B/2	Uniform stochastic Euler, 25, 1.07	$47.9\rightarrow 43.7\rightarrow 30.3$	23.8% $[23.4,24.3]$
SiT-L/2	Uniform stochastic Euler, 25, 1.07	$30.2\rightarrow 26.8\rightarrow 16.1$	23.6% $[23.1,24.0]$
SiT-XL/2	Public hybrid SDE, 50, 1.05	$13.8\rightarrow 12.4\rightarrow 7.7$	22.4% $[21.8,23.0]$
REPA-SiT-XL/2	Public hybrid SDE, 50, 1.05	$10.3\rightarrow 9.2\rightarrow 5.3$	22.7% $[22.1,23.4]$
SiT-XL/2	ODE, 50, 1.05	$11.2\rightarrow 9.7\rightarrow 8.5$	53.9% $[51.4,56.4]$

Appendix

Table 22: Frozen-field FID reduction across samplers, SiT checkpoints, and an ADM U-Net. Each intervention uses 50,000 ImageNet-256 images. Within each row, corresponding images share the class label and initial latent; stochastic rows also share the Brownian path. The FID column gives values as unscaled

\rightarrow

frozen field

\rightarrow

full scaling. The FID-reduction ratio is

(\mathrm{FID}_{\rm unscaled}-\mathrm{FID}_{\rm frozen})/(\mathrm{FID}_{\rm unscaled}-\mathrm{FID}_{\rm full})

. Ratios use unrounded FIDs; displayed FIDs have one decimal place. Brackets give paired 95% intervals for the ratio within each sampler.

Figure 8: Changed-state response components and direction controls for unguided SiT-XL/2 on ImageNet-256 under public hybrid SDE sampling at NFE 50 and gain 1.05. The vertical quantity is the FID reduction from unscaled sampling in FID points, so higher is better; unscaled FID is 13.8. Left: each bar shows a separate intervention, not an additive component. Right: equal-norm parallel and scrambled perpendicular replacements raise FID. Each panel uses a separate population of 50,000 exactly paired images; error bars are paired 95% intervals from 2,000 common-row influence resamples with the real reference fixed. Bar heights and contrasts use unrounded FIDs; raw FIDs elsewhere are displayed to one decimal. Table 24 gives raw values and exact intervals for the direction controls.

Comparator	Realized $\perp$ FID	Comparator FID	FID contrast [95%]
Unscaled	8.5	13.8	$-5.4\;[-5.5,-5.2]$
Norm-matched $\parallel$	8.5	14.7	$-6.2\;[-6.4,-6.1]$
Scrambled $\perp$	8.5	25.3	$-16.8\;[-17.1,-16.6]$

Appendix

Table 24: Response-direction controls for unguided SiT-XL/2 on ImageNet-256 under stochastic sampling. All three blocks use hybrid SDE sampling at constant gain 1.05. The first two use NFE 50; the last uses NFE 25 and 250. Each intervention uses 50,000 images, and lower FID and DINOv2 feature distance are better. Within each comparison, the two settings have exact label, initial-latent, Brownian-stream, and image-row pairing. The controls match the norm of the realized perpendicular velocity change at every step but change its direction. Each contrast is realized perpendicular minus comparator, so negative favors the realized response. Raw DINOv2 values are Fréchet distances in squared feature units and are rounded to whole units; contrasts use unrounded metric values. Brackets give paired 95% intervals from 2,000 common-row influence resamples with the real reference fixed.

Gain	FID $\downarrow$	sFID $\downarrow$	Precision $\uparrow$	Recall $\uparrow$
1.000 (unscaled)	28.0	26.2	0.565	0.630
1.072	12.7	8.3	0.653	0.649
1.100	10.0	5.6	0.675	0.639
1.150 (lowest-FID gain)	8.7	6.7	0.685	0.622

Appendix

Table 25: Image metrics at four constant gains for unguided SiT-XL/2 with uniform stochastic Euler sampling at NFE 25 and 50,000 exactly paired ImageNet-256 images per gain. Gain 1.000 is unscaled and gain 1.150 is the lowest-FID gain on the tested grid. Each contrast is the value at the higher gain minus the value at the lower gain and is computed from unrounded metric values. Lower FID and sFID are better; higher precision and recall are better. Brackets are paired 95% intervals from 2,000 common-row first-order influence resamples for FID and sFID or a 20-group delete-one jackknife for precision and recall.

Solver	NFE	SiT $R$ [95%]	Gaussian $R$ [95%]
Euler–Maruyama	25	$1.0628\;[1.0612,1.0643]$	$1.0527\;[1.0526,1.0528]$
Euler–Maruyama	50	$1.0607\;[1.0590,1.0623]$	$1.0543\;[1.0542,1.0543]$
Euler–Maruyama	100	$1.0595\;[1.0577,1.0612]$	$1.0551\;[1.0550,1.0551]$
Euler–Maruyama	250	$1.0574\;[1.0556,1.0594]$	$1.0556\;[1.0555,1.0556]$
Stochastic Heun	2000	$1.0563\;[1.0543,1.0583]$	$1.0559\;[1.0558,1.0559]$

Appendix

Table 26: Relative endpoint power slope for unguided SiT-XL/2 and a matched linear Gaussian SDE; neither larger nor smaller is intrinsically better for image quality.

R=b_{1.05}/b_{1}

is dimensionless, and

R>1

means more variance toward directions with larger held-out real-data variance under gain 1.05 than under unscaled sampling. The signed contrast is SiT

R

minus Gaussian

R

. Euler–Maruyama rows use 1,280 exactly paired endpoints per setting; the separate 1,000-step stochastic-Heun reference (NFE 2,000) uses 1,024 endpoints per setting, and every row uses 1,024 real-data latents over 256 fixed cross-fitted directions. Brackets give 95% intervals from 2,000 paired fixed-basis bootstrap draws.

Figure 9: FID and covariance sharpening beyond lag correction for unguided SiT-XL/2 with uniform stochastic Euler sampling. Each point shows measured FID minus the lowest measured FID at the same NFE; lower is better. The horizontal coordinate is the relative power slope

S_{N}(\gamma)

in Equation 46 . Segments join measured gain settings, and the dashed line marks the target fitted at the best tested NFE-25 gain. Panel (b) enlarges the region near the grid minima. Each FID uses 50,000 images, and each power slope uses 1,280 paired endpoints in 256 cross-fitted directions from 1,024 real-data latents. We estimate the gains that match the sharpening target, and their FIDs, by linear interpolation between tested settings. Horizontal bars show paired 95% intervals from 2,000 trajectory resamples, conditional on the real-data directions, lag-based gains, and FID grids. They do not include FID uncertainty or uncertainty in locating the best gain between tested settings.

Figure 10: FID across constant gains for the official decoder-free CIFAR-10 1-Rectified Flow model; lower is better and open markers show the lowest measured FID. Panel (a) compares Euler NFE 250 with the official adaptive RK45 solver. Panel (b) uses the velocity-derived SDE in Equation 27 . Each point uses 50,000 paired images within its solver and NFE. All six curves select gain one.

Solver	Gain	FID $\downarrow$	$\Delta$ FID [95%]
Euler NFE 250	0.950	21.7	$+18.7$
	0.975	5.9	$+3.0\;[+2.8,+3.1]$
	1.000	3.0	$+0.0$
	1.025	8.3	$+5.3\;[+5.2,+5.4]$
	1.050	16.6	$+13.6$
	1.075	25.3	$+22.3$

Appendix

Table 27: Decoder-free CIFAR-10 1-Rectified Flow gain curves under Euler NFE 250 and the official adaptive RK45 sampler. FID uses the official TF-Hub Inception network and CIFAR-10 reference; lower is better. The signed contrast is

\Delta\mathrm{FID}=\mathrm{FID}(\text{gain})-\mathrm{FID}(1)

, so a positive value favors identity. Contrasts are computed from unrounded FID values; displayed FIDs use one decimal place. Each gain setting contains 50,000 images. For the two gains neighboring identity, brackets give paired 95% intervals from 2,000 common-row FID influence resamples with the real reference fixed.

CFG $\omega$	$c_{\rm pool}$ [95%]	Pointwise coefficient [95%]	Lowest-FID gain
1.75	0.0258 [0.0252, 0.0264]	0.0251 [0.0245, 0.0256]	1.000
2.00	0.0279 [0.0272, 0.0286]	0.0271 [0.0265, 0.0276]	0.975
2.25	0.0298 [0.0292, 0.0305]	0.0289 [0.0283, 0.0294]	0.950

Appendix

Table 28: Fixed-state classifier-free-guidance field projection for SiT-XL/2 under the public hybrid SDE sampler at NFE 50 and gain one. Each row contains 1,024 trajectories. The pooled coefficient is

c_{\rm pool}=\sum_{t}\Delta t\langle v_{\omega},\Delta\rangle/\sum_{t}\Delta t\lVert v_{\omega}\rVert^{2}

; a magnitude-only account predicts

d\gamma^{\star}/d\omega\approx-c_{\rm pool}

. The pointwise coefficient weights model calls equally; the lower block gives the pooled norm ratio and cosine that form the projection. No quantity has a favorable direction. Brackets give 95% intervals from 2,000 whole-trajectory bootstrap resamples.

Guidance	Lag-based gain [95%]	Unscaled lag RMS	Scalar lag RMS	Lowest-FID gain
1.50	$1.030\;[1.029,1.031]$	9.8	2.2	—
1.75	$1.027\;[1.025,1.028]$	8.6	1.5	1.000
2.00	$1.024\;[1.021,1.028]$	7.6	2.1	0.975
2.25	$1.021\;[1.018,1.025]$	7.2	3.5	0.950
2.50	$1.017\;[1.014,1.018]$	7.2	4.9	—
4.00	$1.003\;[1.000,1.010]$	14.8	14.7	—

Appendix

Table 29: Direct scalar population time lag calibration for guided SiT-XL/2 under the public hybrid SDE sampler at NFE 50. Each candidate uses 1,280 paired paths and all 50 post-update states, including the final Mean step. Lag is RMS population probe-time mismatch multiplied by

10^{3}

; lower is better. Gain brackets are paired 95% intervals from 2,000 whole-trajectory draws with the trained probe and genuine baseline fixed. The same paths select and evaluate each scalar; the genuine baseline remains the class-conditional training path. Lowest-FID gains come from the existing shared 50,000-image grid at guidance 1.75, 2, and 2.25; other entries are not reported (—).

CFG $\omega$	Precision gain 1	Precision gain 0.95	$\Delta$ precision [95%]
1.75	0.845	0.797	$-0.047\;[-0.051,-0.044]$
2.00	0.871	0.837	$-0.035\;[-0.038,-0.032]$
2.25	0.888	0.862	$-0.026\;[-0.028,-0.024]$

Appendix

Table 30: Precision and recall when reducing constant gain from 1 to 0.95 under classifier-free guidance with the SiT-XL/2 public hybrid SDE sampler at NFE 50. Each gain setting contains 50,000 ImageNet-256 images paired by label, initial latent, Brownian stream, and image row; both metrics lie on the zero-to-one scale and higher is better. Each signed contrast is the gain-0.95 value minus the gain-1 value and is computed from unrounded metric values. Brackets give paired normal 95% jackknife intervals from twenty common 2,500-row delete groups with the fixed 10,000-image real reference.

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