Finite-Step Geometry of Marginal-Preserving Velocity Fields in Generative Flows

Ewsey Obzherin; Valentin Khrulkov

cs.LGOct 27, 2026

Finite-Step Geometry of Marginal-Preserving Velocity Fields in Generative Flows

Authors: Ewsey Obzherin, Valentin Khrulkov

Organizations: FusionBrain Lab · MIRIAI

Abstract

Generative flows can admit different velocity fields that induce the same continuous marginal evolution but behave differently after numerical discretization. We study this finite-step non-equivalence while keeping the marginal path and numerical method fixed. Projected modified equations characterize the component of local integration error that changes the marginal law and give sufficient conditions for higher marginal convergence order. For affine fixed-shape distributions, we characterize exact finite-step reachability and derive spectral obstructions and sharp nearest-reachable Wasserstein errors. For diffusion probability-flow ODEs, we construct skew-score perturbations that preserve the exact marginal path under exact scores and quantify their leakage under score error. Controlled Gaussian and mixture experiments recover the predicted order, reachability, and obstruction phenomena.

Figures & tables

Panel of Figure 1 — Figure 1: Rotation can repair an under-expanding Euler step. In this exact two-dimensional Gaussian example ( $\beta h=0.7$ ), the radial step falls short (left). Adding rotation preserves the continuous distribution and gives Euler the correct endpoint radius (middle), while individual particle endpoints remain different. The line–circle intersection on the right gives the required rotation. Circles are Gaussian density contours, not a bounded support. These are analytic illustrations, not benchmark measurements.

Panel of Figure 2 — Figure 2: A correction can change the output without reaching every target. The first two panels show all Euler covariance responses in two dimensions as the rotation $K=\omega J$ varies ( $h=1$ ). With unequal growth rates the response is a parabola; with equal growth rates it is a ray. Targets on these curves are reachable; the displayed targets off them are not. The coordinates are ${\xi_{\rm mix}}=\Gamma_{12}$ and ${\zeta_{\rm exp}}=\tfrac{1}{2}\operatorname{tr}(\Gamma-\Gamma(0))$ , not an isometric Wasserstein chart. Right: target radii $|a|\leq y_{1}<y_{2}$ must share the reachable radius $\varrho_{\star}=(y_{1}+y_{2})/2$ ; the displayed residual cost is exact.

Figure 3: Sampling variance and an energy penalty can jointly suppress a useful correction. Left: the limiting squared Gaussian rotation frequency versus

\tau_{\rm batch}=\lim(n_{\rm b}h^{2})^{-1}

for three

\ell_{\rm pen}=\lim\lambda_{h}/h^{4}

values, with cap

\Omega=2

. Right: at

h=.1

, either effect alone prefers a nonzero correction, while their combination prefers zero. Curves subtract the objective at zero and divide by unpenalized population

C_{h}(0)

. These are exact fixed-feature calculations, with no neural measurements or sampling intervals.

CIFAR-10 / calls	Correction	Test $n$	Base FID	Field FID	$\Delta$ FID
Conditional VP / 4	FH, $\alpha=.75$	50k	67.4914	$\mathbf{63.0786}$	$\mathbf{-4.4128}$
Conditional VP / 8	FH, $\alpha=1$	50k	22.2009	$\mathbf{21.4159}$	$\mathbf{-0.7850}$
Conditional VE / 8	Frozen late	50k	21.4683	$\mathbf{19.1866}$	$\mathbf{-2.2817}$
Unconditional VP / 8	Frozen late	50k	28.9744	$\mathbf{26.5503}$	$\mathbf{-2.4241}$
Unconditional VP / 4	Frozen late	50k	95.3793	97.2882	$+1.9089$
Unconditional VP / 4	Native late, five refits	20k	96.2418	$\mathbf{91.8783\text{--}94.4997}$	$\mathbf{-4.3635\ldots-1.7421}$

Table 1: CIFAR-10 gains depend on the field. Point

\Delta\mathrm{FID}=

field–Base; bold marks lower FID within a paired bank. The native range spans all five refits, not a confidence interval. Rows come from separate studies/banks. Full conditions, adverse/null outcomes and comparator-specific intervals: Sections G.3 , G.5 and G.6 .

Figure 4: Fitting in one time window, then transferring to the other. Top: mean selected

A_{\sigma}

norm at Euler nodes and minimum–maximum range of five independent native refits, including zero amplitudes. Bottom: transferred minus native FID by refit quantile, with conditional pointwise 95% paired-bootstrap bands for each fitted field. Top bands describe calibration/refit variation; bottom bands exclude calibration uncertainty. Negative differences favor transfer; positive differences favor the native window.

Symbols	Meaning
$\rho_{t}$ , $\widetilde{\rho}_{t}$ , $\rho_{t}^{\rm B},\rho_{t}^{\rm G}$	Prescribed density, comparison-flow density, and actual learned Base/Gauge densities.
$v,u,w=v+u$	Base velocity, admissible correction, and selected representative.
$\Pi_{\rho}$ , $\Psi_{h}^{w}$ , $\mathfrak{R}_{t,h}$ , $\mathcal{M}_{t,h}$	Weighted gradient projection, numerical step map, response law, and reachable set.
$m_{t}$ , $\Sigma_{t}$ , $\Sigma_{\star}$	Center (mean when integrable), scatter (covariance when normalized), and whitened target scatter.
$Z$ , $P_{\rm PCA}$	Spherical/Gaussian random input and fixed calibration PCA basis.
$B$ , $\widehat{B}=L^{-1}BL$ , $S$ , $K$ , $J$	Affine generator, whitened generator, symmetric strain, skew control, and orthogonal complex structure.

Table 2: Recurring notation. Definitions and assumptions remain in the indicated text; the table is a reading aid.

Theory	Prediction and retained check	Location
Projected marginal order	E0: trajectory versus law convergence	Section A.5
Affine reachability	E0/E1: parity, spectral and rank/cap obstructions	Section B.3
Solver response	E0/E3: exact roots, native fitting and solver/grid transfer	Sections C.2 and C.4
Euclidean and radial limits	Stationary moment defect, torus contrast and shellwise regularity	Appendices D and D.2
Mixture response	E2: separated/overlap outcomes and curvature counterexample	Sections D.3 and D.4

Table 3: From theoretical predictions to controlled checks. Each proof cluster is followed by its corresponding verification; pretrained evidence remains separate in Appendix G .

Appendix figures & tables41 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Controlled marginal-order check. Trajectory and law-error curves isolate the marginal-order prediction. Solver-root and curvature diagnostics accompany their corresponding proof blocks. This deterministic fixture has no sampling interval.

Check	Retained result and boundary
E0: parity	Even-dimensional $W_{2}\leq 1.2\!\times\!10^{-7}$ ; odd-formula error $<4\!\times\!10^{-14}$ .
E1: spectra, anisotropy	Pairing-cost median/max gap $2.2\!\times\!10^{-11}/1.4\!\times\!10^{-7}$ ; at directional violation 0.724, optimized cost 0.0791 exceeds lower bound 0.0656.

Appendix

Table 4: Controlled affine reachability results.

Figure 6: Finite-step reachability and spectral obstruction across the retained dimensions and fixtures. Even exactness, odd residuals and paired spectral costs are distinct tests. Floating-point display floors do not turn numerical zeros into positive physical errors; fixture ranges are descriptive.

Figure 7: All four repeated/simple-strain controls at 81 recorded times. These deterministic covariance and law-discrepancy curves verify the commuting whole-path boundary in Proposition 24 , without confidence intervals.

Figure 8: Controlled solver-response check. The recorded roots accompany Corollary 26 . They are deterministic roots, not sampling intervals or matched-cost comparisons.

Check	Retained result and boundary
E3: Digits, $d=16$ , N4	Class-scale RMSE Base/LO/FH/target: 0.46115/0.17415/0.02383/0.02623; split-bootstrap Base–FH improvement 0.43732 $[0.43374,0.44137]$ , positive in all ten refits.
E3: solver/grid transfer	Two-step Heun scale error Base/Euler-FH/Heun-FH: 0.13366/0.77290/0.02522. Jittered Euler Base/transferred/recomputed: 0.65885/0.05061/0.02735. This matches the number of partitions, not the network-call budget.

Appendix

Table 5: Exact latent-model integration and transfer results. Target-model sampling and solver-transfer errors are distinct.

Figure 9: Data-derived exact latent Digits models: dimensions 8/16/32 over integration budgets, followed by receiving-solver and grid transfer. Panels (a)–(d) use supplied 95% split-bootstrap intervals after averaging three particle seeds within each of ten independent refits; solver/grid panels use descriptive ranges over 12 fixed seeds. These are surrogate-model integration errors, not true-image-law guarantees or matched-call solver comparisons.

Figure 10: Cached exact-latent Digits, $d=16$ , Euler N4, replicate 0, source RNG seed 16000. Base/shuffled/LO/FH share initial latents, while the exact target is an independent draw with the same labels (RNG seed 32000). Linear PCA decoding illustrates the surrogate model in Section C.4 ; it is separate from pretrained neural sampling.

Figure 11: Archived radial CDF curves for the cubic and quintic full-law fixtures. The cubic field is

C^{1}

, not

C^{2}

, at the origin; the quintic counterpart is smooth. Covariance matching alone does not settle full-law transport. These are point diagnostics without confidence intervals.

Check	Retained result and boundary
E2: hard GMM, N4	Component-scale RMSE Base/energy-random/analytic FH/target: 0.393/1.096/0.059/0.056; mode-mass errors remain comparable to target sampling.
E2: overlap stress, N4	Analytic FH 0.0875 $\pm$ 0.0033; calibrated FH 0.0887 $\pm$ 0.0033; curvature-energy 0.0929 $\pm$ 0.0021 (eight seeds). Calibration does not improve the analytic field; curvature is competitive.

Appendix

Table 6: Separated and overlapping mixture outcomes. All negative and competing-control results are retained.

Figure 12: Held-out mixture checks across actual integration budgets. Hard and overlapping mixtures retain all matched controls and negative outcomes, including competitive curvature and the failure of extra calibration to improve on analytic FH. Panels (a)–(c) show descriptive ranges over four seeds; (d) shows supplied mean

\pm

SE over five seeds; (e)–(f) show ranges over five seeds. These bands are separate from neural paired-FID intervals.

Figure 13: Controlled curvature counterexample. Large full acceleration coexists with negligible projected response. This deterministic fixture has no sampling interval.

Figure 14: Gaussian calibration optima. Left: optimal frequency for unpenalized population, expected iid minibatch (

n_{\rm b}=128

), and penalized population (

\lambda=10^{-5}

) objectives. Middle/right: unpenalized curves at

h=.10,.30

, subtracting each curve’s value at zero and dividing by

C_{h}(0)

. Formula curves use fixed unit scales and an exact Base reference; they are not neural measurements.

Family	Update/scale bank	Adam updates, rate; $\lambda_{\rm pen}$	Rank/cap/profile
Primary full-cloud FH; later checkpoint/refit FH; rank/cap	Full 4096, full-node mean $N^{-1}\sum_{k}\mathcal{L}_{k}$ ; full-node scale. Matched projected-LO FH uses 3072.	800, .012; $10^{-5}$	$r=2$ at N8, 8 at N4/N16/N20; cap .5. Rank study $r=1,2,4,8,16$ , caps .25/.5; eight global log-noise knots.
Early actual-Heun diagnostic	Cyclic nodes, 128 without replacement from 4096/node; batch scale; 64 Gaussian directions only.	400, .008; 0	$r=2/8$ at N8/N16, cap .5; eight global knots.
Matched Euler/Heun cross-fit	Cyclic eight nodes, 128 without replacement from 4096; batch scale.	800, .008; 0	$r=2$ , cap .5; eight global knots; identical fit seeds within solver pair.
Native early/late windows	Cyclic active half only, 128 without replacement from 4096; batch scale.	800 active updates, .008; 0	$r=8/2$ for N4/N8, cap .5; actual-node masked coefficients.
RF native N8/N16	Cyclic nodes, 128 with replacement from 4096; full-node scale fixed beforehand.	800, .008; $10^{-5}$	$r=2$ , $\\|K_{t}^{\rm RF}\\|\leq.5$ ; eight linear-time knots, not the EDM cap.

Appendix

Table 7: Fitting families. “Full” uses all listed calibration particles/nodes per update; “cyclic” uses one node. Unless stated otherwise,

P_{\rm PCA}

has 32 columns and the map has 128 directions.

r

counts rank pairs; calibration sizes differ from test sizes.

Component	Recorded scope and limitation
Coarse sampling	Actual network calls are reported with each solver comparison. Worker generation time excludes fitting, feature extraction, I/O and scheduling.
Offline selection	Calibration, PCA, fitting, dense references and six validation candidates are additional costs. No controlled end-to-end deployment or break-even measurement is available.
Diagnostic timing	Elapsed local-shard hours include feature calls/I/O; calibration-cloud wall seconds and sampling worker-hours are distinct and cannot be added.

Appendix

Table 8: Cost accounting and its limits. Detailed recorded timings are in the code supplement; heterogeneous execution timings do not establish latency gains or amortization.

Study/scope	N	FID or $\Delta$ FID	Uncertainty/interpretation
Initial fixed fields, 50k	8	Base: $22.4244$ ; Random: $21.9074$ Weak LO: $22.0634$ ; FH: $21.6174$	FID point values
Full amplitude $\alpha=1$ , 50k	4	Base: $67.653$ ; FH: $78.470$ Weak LO: $62.435$ ; Random: $70.647$	FH worsens FID despite lower KID
28 fixed-field seed blocks	8	$\bm{-0.80831}$	$t$ 95 $[-0.81772,-0.79890]$
28 fixed-field seed blocks	16	$\bm{-0.03682}$	$t$ 95 $[-0.04009,-0.03354]$
28 fixed-field seed blocks	20	$\bm{-0.01890}$	$t$ 95 $[-0.02195,-0.01586]$
Frozen validation choice, 50k	4	$\bm{-4.41279}$ ( $\alpha=0.75$ )	Paired95 $[-4.68740,-4.22669]$

Appendix

Table 9: Initial conditional-CIFAR VP results (upper block), distinct from later conditional VE. Differences are FH–Base unless the comparator is explicitly named. Fixed-field means and

t

95 intervals concern 28 matched 50k sampling blocks, not 28 fits. Refit means/

t

95 concern eight independent calibration/PCA/optimizer runs on one common 20k bank. The eight N4 full pipelines instead use separate calibration, validation and fresh 50k test banks per pipeline; their range is descriptive and intervals conditional per field. All fixed-field methods and adverse/null outcomes remain. Bold denotes active improvements under the stated uncertainty, not calibration-population coverage.

Checkpoint/N	Refit scope	$\alpha$	$\Delta$ FID/range	Improve/worsen	Below/above
CIFAR VE/8	five refits	$1$	$\bm{[-0.7555,-0.4993]}$	5/0	5/0
CIFAR uncond./8	five refits	$1$	$\bm{[-0.8935,-0.6967]}$	5/0	5/0
CIFAR uncond./4	five refits	$0.75$	$[-1.0762,+0.5071]$	3/2	2/1
FFHQ/4	five refits	$1$	$\bm{[-0.2814,-0.2001]}$	5/0	5/0
FFHQ/8	five refits	$1$	$[+0.0117,+0.0161]$	0/5	0/4
AFHQ/8: every independent refit, including the three disabled corrections

Appendix

Table 10: All 25 Phase5 refits plus all five AFHQ refits on common 20k test banks per checkpoint/NFE. Upper rows give five-refit

\Delta

FID ranges, point improve/worsen counts and counts of conditional pointwise 95% intervals below/above zero (not refit-population coverage). Lower rows give individual

\Delta

FID and those intervals. Bold marks active all-five improvements with five below-zero intervals, or an individual active below-zero interval.

\alpha=0

disables the gauge: its residual signs, including a positive interval, are backend observations rather than learned effects.

Figure 15: All 30 Phase 5/6 independent refits, ordered within each prespecified family by observed FID effect; the separate panel retains all eight conditional-CIFAR N4 fresh-50k pipelines. Bands are the existing conditional intervals for each fitted field, not confidence bands for a refit population or quantile trend. Open symbols retain

\alpha=0

; their tiny nonbitwise residuals are not learned effects.

Figure 16: AFHQv2 VP, Euler N8: Base, original FH and all five native refits on the same first eight seeds of the common 20k test bank. Refits 1–3 select $\alpha=0$ ; any tiny backend pixel residual is not a learned-gauge effect. The common appearance does not establish marginal equivalence.

Comparison	Recorded result	Scope/uncertainty
Projected LO–matched FH,8 calls	$+0.07929$	Conditional paired95 $[+0.01858,+0.13887]$
Projected LO–matched FH,16 calls	$-0.00337$	Conditional paired95 $[-0.03822,+0.03062]$
Prespecified advancement	$\|\Delta\mathrm{FID}\|\geq.10$ :FAIL	Direction-agnostic pilot gate; no gated50k result
16 time-knot shuffles vs FH	$[+0.11726,+0.62612]$	0/16/0 intervals below/above/containing zero; 64 resamples
16 spectrum orientations vs FH	$[-0.08190,+0.02965]$	1/0/15 intervals below/above/containing zero; 64 resamples
Log-grid N8: transferred/native fit	$22.2376/22.3275$ FID	Matched20k point values; no direct interval

Appendix

Table 11: Original conditional-CIFAR VP projected-LO, profile/orientation, grid and call-context tests. Unless marked FID, numbers are method–named-reference

\Delta

FID. Pilot/draw ranges are descriptive; original 32 control intervals use 64 bootstrap resamples, distinct from the 128-replicate later contrasts. Orientation results preserve spectra without state-conditional energy matching. The screened fresh 50k one-call comparison is exploratory; it finds no measured integer-call saving.

Checkpoint/N	$\alpha$	$\Delta$ FID	Conditional 95%	Own SW2: Base $\to$ FH
CIFAR VE/4	$0.75$	$\bm{-5.52007}$	$[-5.68471,-5.29807]$	$0.124881\to 0.107342$
CIFAR VE/8	$1$	$\bm{-0.67896}$	$[-0.72016,-0.64694]$	$0.067337\to 0.065565$
CIFAR VE/16	$1$	$\bm{-0.03520}$	$[-0.04953,-0.01744]$	$0.035883\to 0.035726$
CIFAR uncond./4	$0.75$	$+0.40445$	$[+0.11676,+0.65074]$	$0.160136\to 0.147643$
CIFAR uncond./8	$1$	$\bm{-0.83403}$	$[-0.88015,-0.77928]$	$0.084550\to 0.082423$
CIFAR uncond./16	$1$	$\bm{-0.03262}$	$[-0.04973,-0.01568]$	$0.040564\to 0.040390$

Appendix

Table 12: Nine final checkpoint/NFE cells: FID uses fresh 50k banks and own-SW2 uses 20k banks. FH minus Base FID has conditional pointwise 95% paired intervals, alongside feature SW2 to each field’s own Heun 256 flow (point measurements). Bold denotes below-zero active FID intervals, not a confidence statement about SW2. All own distances decrease (0.01%–14.04%), including two adverse FID cells; the FFHQ16 FID interval includes zero.

Figure 17: All nine fresh-50k checkpoint/NFE cells. Real-data FID contrasts have the registered conditional pointwise 95% paired intervals; empirical own-field distances have no population interval. Lower own-field integration error can coexist with worse FID, notably at unconditional CIFAR N4.

Diagnostic	Recorded result	Scope/uncertainty
AFHQ refit 0	Own SW2 $.04854216\to.04857189$	Point only; lower point FID, higher own-error
Independent BaseA–BaseB	VE $\approx.005750$ ; uncond. $\approx.005641$ ; FFHQ $\approx.005279$	Same-metric sampling references
Coupled Heun 256–512	SW2 $.000415$ – $.000709$	Same-noise numerical refinement
Independent dense kernel test	All Holm-adjusted $p=1$	Fixed 4096 subsets; 512 permutations; six amplitudes
Clipped 128-function mean bound	$\approx.0771$ – $.0776$ ; empirical max $.0102$ – $.0107$	Simultaneous finite-map means, not FID/SW2
PRDC $k=5$ : eight N4 pipelines	Recall $\uparrow$ ;precision/density/coverage $\downarrow$ in all eight	Three paired 5k draws;descriptive signs

Appendix

Table 13: Own-error counterexample and all reference/leakage/diversity diagnostic families. Refinement distances are neither independent floors nor certified reference errors, population intervals or equality tests. Non-rejection and finite-map bounds do not establish marginal equality; mixed PRDC changes do not prove absence of mode collapse. No row is highlighted as evidence of a universal benefit.

Figure 18: All 20 CIFAR rank/cap fields, with PRDC precision, recall, density and coverage. Curves average three paired 5k subsamples and scale changes by 100 (percentage points for precision/recall/coverage, dimensionless density times 100); shading is their descriptive minimum–maximum range, not a confidence interval or independent refit uncertainty. Mixed diversity changes do not prove absence of mode collapse; FFHQ raw-real-feature PRDC is unavailable.

Condition	CIFAR VE/8	CIFAR uncond./8	CIFAR uncond./4	FFHQ/8
Base	$+0.6866$ $[+0.6505,+0.7138]$	$+0.8838$ $[+0.8404,+0.9389]$	$-0.5127$ $[-0.8195,-0.2143]$	$-0.0107$ $[-0.0190,-0.0029]$
FH	$0$ (reference)	$0$ (reference)	$0$ (reference)	$0$ (reference)
Reverse	$+0.6778$ $[+0.6501,+0.7047]$	$+0.8555$ $[+0.8244,+0.8917]$	$\bm{-1.3877}$ $[-1.6944,-1.0814]$	$\bm{-0.0110}$ $[-0.0209,-0.0004]$
Flat	$+0.6777$ $[+0.6467,+0.7056]$	$+0.8580$ $[+0.8238,+0.9015]$	$\bm{-0.7105}$ $[-1.0159,-0.4280]$	$\bm{-0.0079}$ $[-0.0158,-0.0005]$
Early	$+0.6519$ $[+0.6249,+0.6829]$	$+0.8130$ $[+0.7843,+0.8548]$	$\bm{-1.0661}$ $[-1.3705,-0.7692]$	$\bm{-0.0128}$ $[-0.0218,-0.0031]$
Late	$\bm{-1.5950}$ $[-1.6315,-1.5579]$	$\bm{-1.5403}$ $[-1.5815,-1.4895]$	$+1.3962$ $[+1.3100,+1.4784]$	$+0.1014$ $[+0.0889,+0.1173]$

Appendix

Table 14: All 24 fresh 50k confirmations, fixed before viewing pilot signs. Each cell is condition minus original FH FID and its conditional pointwise 95% paired interval. Base intervals negate the registered FH–Base intervals; original FH is the zero reference. Bold marks active profile changes with intervals wholly below zero, not a universal temporal-position or familywise claim. Energy-matched profiles and spectrum-preserving orientations are distinct controls.

Figure 19: Descriptive empirical distributions of eight time shuffles/eight orientations per cell. Shading is the descriptive middle 50% of the eight draws, not a confidence interval. Curves are fixed-field pilot controls, not

p

-values; fresh 50k confirmations are tabulated in Table 14 . Profile-energy matching and spectrum-preserving orientation controls are different interventions.

Figure 20: All 24 fresh-50k frozen-profile confirmations, with the registered conditional pointwise paired-FID intervals. Control labels are categorical, so they are not joined as a time curve. Late redistribution helps the two CIFAR N8 cells but harms unconditional N4 and FFHQ N8. This frozen-field intervention is separate from native-window retraining in Fig. 4 .

Panel of Figure 21 — Figure 21: Frozen-profile confirmation: conditional VE (top) and unconditional VP (bottom) CIFAR-10, Euler N8. Within each panel, Base/FH/early/late rows share the eight displayed per-image seeds and any conditional labels from the fresh 50k bank. Early/late redistribute the existing fitted profile; source $\alpha$ names its original selected strength, not a new native-window fit.

Checkpoint/N	Window	Native–Base	Transfer–native	$\alpha=0$	Below/above
CIFAR VE/8	early	$[-0.0018,+0.0012]$	$[+0.0000,+0.0020]$	2	0/0
CIFAR VE/8	late	$[-0.0432,+0.0015]$	$[+0.0003,+0.0511]$	0	0/1
CIFAR uncond./8	early	$[-0.0047,+0.0012]$	$[+0.0004,+0.0040]$	0	0/0
CIFAR uncond./8	late	$[-0.0486,+0.0004]$	$[-0.0179,+0.0430]$	1	0/0
CIFAR uncond./4	early	$[-0.0079,+0.0026]$	$[-0.0065,+0.0082]$	2	1/0
CIFAR uncond./4	late	$\bm{[-4.3635,-1.7421]}$	$[+0.5902,+2.4492]$	0	0/5

Appendix

Table 15: All native-window cells, five refits per row. Native–Base and transferred–native ranges retain all point estimates (four-decimal rounding); zero selections are counted explicitly. The last column counts direct-transfer conditional 95% intervals below/above zero, not simultaneous calibration coverage. Bold marks active all-five native gains or active all-five transfer gains; the latter has only three below-zero intervals. Zero-amplitude backend residuals are not learned effects.

Panel of Figure 22 — Figure 22: Native-window refit 0: conditional VE (top) and unconditional VP (bottom) CIFAR N8. Arrows give fitting window $\to$ receiving window. Both native and transferred rows are retained; transfer inherits the native-selected amplitude without receiver tuning. Refit 0 is chosen by index uniformly across cells, not by appearance or FID.

Fit $\to$ receiver	CIFAR VE	CIFAR uncond.	FFHQ
Euler $\to$ Euler (8 calls)	$\bm{-0.04242}$ $[-0.06876,-0.02280]$ $\alpha=1$	$\bm{-0.03922}$ $[-0.06648,-0.01572]$ $\alpha=1$	$-0.00169$ $[-0.00477,+0.00194]$ $\alpha=0.125$
Euler $\to$ Heun (15 calls)	$+0.09914$ $[+0.09076,+0.10550]$ $\alpha=1$	$+0.05597$ $[+0.05007,+0.06246]$ $\alpha=1$	$+0.00079$ $[-0.00129,+0.00369]$ $\alpha=0.125$
Heun $\to$ Euler (8 calls)	$-0.00050$ $[-0.00813,+0.00690]$ $\alpha=0.125$	$+0.00015$ $[-0.00015,+0.00045]$ $\alpha=0$ (off)	$+0.00213$ $[-0.00269,+0.00807]$ $\alpha=0.25$
Heun $\to$ Heun (15 calls)	$-0.00161$ $[-0.00424,+0.00057]$ $\alpha=0.125$	$+0.00000$ $[-0.00008,+0.00033]$ $\alpha=0$ (off)	$\bm{-0.00311}$ $[-0.00573,-0.00030]$ $\alpha=0.25$

Appendix

Table 16: All 12 fitted-field solver contrasts on common 20k banks. Each cell gives field minus the receiving solver’s Base FID, its conditional pointwise 95% paired-bootstrap interval, and the inherited native-selected amplitude. The six receiving Base rows have

\Delta=0

. Bold denotes active fields with intervals wholly below zero;

\alpha=0

rows are recorded backend residuals. These are comparisons to receiving Base, not direct transfer–native intervals.

Figure 23: All 18 native-fit/receiving-solver conditions, retaining disabled corrections. Existing conditional pointwise 95% paired intervals compare each field with the receiving solver's own Base. Transferring an Euler-native correction can worsen Heun. Receiver costs are 8 and 15 network calls, so this matrix does not make an equal-cost claim.

Checkpoint	Cap	$\Delta$ FID range	Active/off	Below/above/ contains zero	Own lower
CIFAR VE	$0.25$	$\bm{[-0.1198,-0.0642]}$	5/0	5/0/0	5/5
CIFAR VE	$0.5$	$\bm{[-0.7676,-0.6219]}$	5/0	5/0/0	5/5
CIFAR uncond.	$0.25$	$\bm{[-0.1571,-0.0740]}$	5/0	5/0/0	5/5
CIFAR uncond.	$0.5$	$\bm{[-0.8748,-0.8260]}$	5/0	5/0/0	5/5
FFHQ	$0.25$	$[-0.0170,+0.0000]$	3/2	3/0/2	3/3
FFHQ	$0.5$	$[-0.0084,+0.0147]$	5/0	0/0/5	5/5

Appendix

Table 17: All 30 rank/cap fields: five fixed rank-pair settings

r=1,2,4,8,16

per row, not five refit replications. FID ranges retain every point estimate; below/above/contains-zero counts classify conditional pointwise 95% intervals. The last column counts active fields with smaller point own-SW2, each versus its own Heun 256 reference; no population-SW2 interval is available. Bold marks rows with five active FID improvements and five below-zero intervals. Zero-amplitude residuals remain in the ranges.

Figure 24: Top: change in own-field SW2 across rank pairs and caps, with each field compared against its own Heun 256 reference; no population-SW2 interval is supplied. Bottom: FID change versus Base with conditional pointwise 95% paired-bootstrap bands. Zero amplitudes and independent Base–Base distances remain separate. Rank does not yield monotone FID gains.

Checkpoint/N	Intervals	FH	Frozen early	Frozen late
CIFAR VE/8	8	0/8/0	0/4/4	0/5/3
CIFAR uncond./8	8	0/7/1	0/4/4	0/4/4
CIFAR uncond./4	4	0/3/1	0/2/2	0/3/1
FFHQ/8	8	0/7/1	0/4/4	0/4/4
All non-Base	28 $\times$ 3	$0$ below, $55$ above, $29$ include zero (84 total)

Appendix

Table 18: Local coupled-state RMS minus Base: counts of 512-replicate conditional pointwise 95% intervals below/above/including zero. All 84 non-Base contrasts and 28 Base zero-reference rows are retained. Among non-Base rows, 24 intervals are exactly

[0,0]

. Four above-zero late rows have zero correction at the left Euler node but become active inside the dense interpolated interval; they are not globally inactive-field effects.

Figure 25: Top: local coupled-state RMS error minus the corresponding Base error over all coarse intervals. Bands are 512-replicate conditional pointwise 95% paired bootstraps; all 84 outcomes are counted in Table 18 . Bottom: own-field 32/64-substep RMS audit on the same bank, a numerical-convergence check without an independent floor or confidence interval. Denoised-feature proxy and characteristic errors remain separate.

Field – comparator (calls)	CIFAR VE	CIFAR uncond.	FFHQ
Euler FH8 – Base8 (8/8)	$\bm{-0.7259}$ $[-0.7904,-0.6797]$	$\bm{-0.9069}$ $[-0.9896,-0.8284]$	$+0.0165$ $[+0.0017,+0.0329]$
Euler FH8@16 – Base16 (16/16)	$-0.0259$ $[-0.0523,+0.0027]$	$\bm{-0.0346}$ $[-0.0630,-0.0065]$	$-0.0031$ $[-0.0104,+0.0035]$
Euler FH8 – Heun4 (8/7)	$\bm{-64.6699}$ $[-65.4678,-64.0654]$	$\bm{-71.7132}$ $[-72.4355,-71.0759]$	$\bm{-114.3447}$ $[-115.2294,-113.5988]$
Euler FH8 – Heun5 (8/9)	$\bm{-14.1552}$ $[-14.7566,-13.7185]$	$\bm{-10.0018}$ $[-10.6906,-9.5506]$	$\bm{-28.6022}$ $[-29.2142,-28.1417]$
Euler FH8@16 – Heun8 (16/15)	$+3.3165$ $[+3.1017,+3.5232]$	$+4.5093$ $[+4.2581,+4.6729]$	$+1.3363$ $[+0.9899,+1.6234]$
Euler FH8@16 – Heun9 (16/17)	$+4.6772$ $[+4.4883,+4.8891]$	$+5.7780$ $[+5.5436,+5.9581]$	$+4.4108$ $[+4.0715,+4.6783]$

Appendix

Table 19: All 18 fixed cost-context contrasts from 24 fresh 20k datasets. Each cell is field–comparator

\Delta

FID and its conditional pointwise 95% paired-bootstrap interval; parentheses give actual field/comparator calls. Bold marks active below-zero intervals. FH8@16 carries the frozen N8 field to Euler16. The 8-versus-7/9 and 16-versus-15/17 brackets are approximate, not exactly equal cost or measured call savings.

Figure 26: Top: recorded FID versus actual network calls. Bottom: Euler FH minus the indicated comparator, with conditional pointwise 95% paired-bootstrap bands; these difference intervals are not absolute-FID intervals. Competitiveness changes across budgets; Euler 16 carries the frozen N8 field. Connecting measured points does not imply interpolated call savings.

N	$\alpha$	$\Delta$ FID	Conditional 95%	Own SW2: Base $\to$ FH
8	$0.75$	$-0.00912$	$[-0.03317,+0.01396]$	$0.05707995\to 0.05704312$
16	$0.5$	$-0.00836$	$[-0.02428,+0.00342]$	$0.03542936\to 0.03543288$

Appendix

Table 20: Original independent-path RF pilot: all two held-out 10k FID contrasts and four own-SW2 point measurements. FID intervals are conditional pointwise 95% paired bootstraps; both include zero. Own distances compare each field to its own Heun 256 reference, without an independent floor, population-SW2 interval or equivalence inference. No entry is bolded as an established FID improvement.

Figure 27: Original independent-Gaussian RF pilot: all validation amplitudes, the frozen choices, both fresh-10k FID contrasts and the two own-field comparisons. The last panel shows

(\mathrm{SW2}_{\rm FH,own}-\mathrm{SW2}_{\rm Base,own})\times 10^{5}

; all four original values remain in the scalar record. FID shading is the existing conditional pointwise 95% paired interval, and both intervals span zero. Validation and own-SW2 points have no population interval. One checkpoint and two fits do not establish family-wide robustness or neural marginal preservation.

Figure 28: The original independently coupled RF checkpoint, Base/FH at N8/N16, with validation-selected strengths .75/.5 and the first eight fresh-10k seeds. These matched examples accompany the null paired-FID result in Section G.11 ; visual similarity does not establish marginal equivalence.

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Yonsei University · KAIST

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Oct 21, 2026

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FusionBrain Lab · MIRIAI

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LLM Suite Team, JP Morgan Chase & Co.

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