Is your uncertainty map wrong, or is its target? Exact diagnostics for the Tweedie diagonal, and a gradient-free alternative

Vicent Ribas; Anna Oliveras Tous

cs.AISep 27, 2026

Is your uncertainty map wrong, or is its target? Exact diagnostics for the Tweedie diagonal, and a gradient-free alternative

Authors: Vicent Ribas, Anna Oliveras Tous

Organizations: Eurecat · University of Barcelona

Abstract

A diffusion model can predict a follow-up medical scan from a baseline, but a clinician needs a per-voxel map of where that prediction can be trusted. Many such maps approximate the diagonal of the Tweedie posterior covariance, and are evaluated against another approximation of it, so whether the estimator or the target limits them is unclear. We compute the exact diagonal on six checkpoints across fourteen model-corpus conditions. Hutchinson at M=200 tracks it at rank agreement of at least 0.92 everywhere, yet in four of the fourteen the exact diagonal is anti-correlated with the denoising error, reaching -0.13, so a faithful estimator reproduces that reversal. All four are real-image conditions; on the models' own samples the reversal does not appear, so evaluating on generated samples flatters this family. What limits these maps is the target, not the estimator. We then introduce Tweedie Probe-Tangent (T-PT), a gradient-free residual probe that corrupts one model-supported prediction repeatedly and measures the voxel-wise variance of the denoiser's response. T-PT reads a different functional of the same Jacobian, and its exact second-order form ranks with the diagonal wherever the diagonal reverses; at thirty probes it returns a map too unstable to reproduce that ranking, while Hutchinson at M=5 already reproduces it, so T-PT there is not evidence against the reversal. We offer it as an instrument, not a better approximation. On brain MRI at full resolution, where every Jacobian-based estimator we test runs out of memory, T-PT leads a twenty-chain Monte-Carlo ensemble on five of eight endpoints inside tissue and trails it on none, at 16x fewer network evaluations; over the whole volume the ensemble leads, and fifty chains close the tissue gap. On lung CT the ensemble is ahead throughout. Both lose most of their discrimination where the change is, which remains open.

Figures & tables

Figure 1: Spearman correlation with the denoising error, per condition, mean over

100

images, ordered by the exact diagonal (Table 7 gives the numbers and the intervals, in its own ordering). In the shaded block the exact diagonal is anti -correlated with the error, and Hutchinson at

M{=}200

follows it there as everywhere else, so what fails is the target and not its approximation. A condition is harmful when its interval lies entirely below

-0.02

(dashed), a threshold fixed on the synthetic manifolds of App. I before any natural-image score was read: four of fourteen for the exact diagonal, four for Hutchinson, six for the row-sum, and none for T-PT at

K=30

, whose map runs flat along zero there rather than tracking the diagonal. Sec. 4.4 and App. P show that this flatness is a property of that budget: with

4

–

16\times

the probes T-PT is harmful on the two inet conditions too.

	Lung CT (PNG, $N{=}130$ )				Brain MRI (OASIS, $N{=}211$ )
	T-PT	MC	Hutch.	$\Delta$ [95% CI]	T-PT	MC	$\Delta$ [95% CI]
$\rho$ , full volume	0.669	0.693	0.404	$-0.022\,[-0.037,-0.007]$	0.518	0.559	$-0.038\,[-0.041,-0.035]$
$\rho$ , tissue	0.670	0.694	0.405	$-0.022\,[-0.037,-0.007]$	0.349	0.337	$+0.016\,[+0.006,+0.028]$
$\rho$ , interior	0.612	0.633	0.326	$-0.023\,[-0.043,-0.005]$	0.334	0.318	$+0.022\,[+0.009,+0.037]$
$\rho$ , edges	0.352	0.528	0.177	$-0.147\,[-0.186,-0.110]$	0.248	0.245	$+0.003\,[-0.005,+0.010]$
$\rho$ , change region	0.100	0.265	0.033	$-0.134\,[-0.183,-0.085]$	0.128	0.133	$-0.003\,[-0.013,+0.008]$

Table 1: Clinical results against the common error map. Estimator columns are per-volume means;

\Delta

is T-PT minus MC-DDIM, so on the two rows marked

\downarrow

a positive

\Delta

favours MC-DDIM, with its paired

95\%

confidence interval resampling participants (

63

on OASIS,

47

on lung CT, where the full volume is

99.5\%

tissue). Hutch. is Hutchinson at

M{=}200

on the probe’s own chain, which has no brain column because at

128^{3}

it exhausts memory (Sec. 5.3 ); on lung CT, the probe’s chain, the reference and the error map share one conditioning (App. M ). Bold marks the better estimator where the interval excludes zero. Volume-weighted contrasts: Table 15 .

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Schedule-induced amplification in the additive variance recursion. Left: cumulative gain $\prod a_{s}^{2}=1/\bar{\alpha}_{t}$ from starting accumulation at step $t$ ; a one-per-cent budget gives $s_{\kappa}=16$ . Right: accumulated variance with no mask, a late mask ( $t{=}64$ ), and the schedule-calibrated mask. The mask bounds inflation but cannot make the recursion contractive. Every coefficient comes from the schedule; no model or data is involved.

$h$	$h/\sigma$	predicted	measured	rel. error
$-0.20$	$-2.0$	$1.5625$	$1.5425$	$-1.28\%$
$-0.10$	$-1.0$	$1.2346$	$1.2206$	$-1.13\%$
$\phantom{-}0.00$	$0.0$	$1.0000$	$0.9898$	$-1.02\%$
$+0.10$	$+1.0$	$0.8264$	$0.8188$	$-0.92\%$
$+0.20$	$+2.0$	$0.6944$	$0.6886$	$-0.84\%$

Appendix

Table 2: Tangent eigenvalue of

\operatorname{Cov}(r)

against normal offset, at

\sigma=0.10

; “predicted” is Eq. ( 23 ). The prediction moves over a factor of

2.3

and the measurement follows it to about one per cent throughout, with the sign the theorem requires: probing inside the circle inflates the eigenvalue above one, outside deflates it, and

h=0

recovers the flat result. The normal component stays below

1.3\times 10^{-2}

, confirming that the normal direction is identified as the decomposition claims.

Stage	Manifold	$\dim\mathcal{M}$	codim
S1	$S^{1}$ , one Gaussian blob on a circular orbit	$1$	$1023$
S2	$T^{2}$ , two independent blobs	$2$	$1022$

Appendix

Table 3: The two synthetic regimes. Codimension is severe by design: a rank-

1

covariance inside

\mathbb{R}^{1024}

is not something a normalisation artefact produces by accident.

	$\lambda_{1}$	$\lambda_{2}$	$\lambda_{3}$	subspace alignment with $T_{x_{0}}\mathcal{M}$
S1 ( $\dim=1$ )	$0.9931$	$0.0118$	—	$0.9998$
S2 ( $\dim=2$ )	$1.052$	$0.839$	$0.018$	$0.9967$
Prop. 1 (broad-prior limit)	$1$	$1$ if $\dim\geq 2$ , else $0$	$0$	$1$

Appendix

Table 4: Spectrum and leading eigenspace of

\operatorname{Cov}(r)

against the analytic prediction. Alignment is the mean squared canonical correlation between the leading

\dim\mathcal{M}

eigenvectors and the analytic tangent span. On S1 the eigengap

\lambda_{1}/\lambda_{2}

84.2

; on S2 the gap between the second and third eigenvalues is a factor of

47

, so the intrinsic dimension is read off the spectrum without a threshold being tuned.

estimator	peak memory	wall time	one $128^{3}$ volume
T-PT , $K{=}30$	$33.8$ GiB	$66.7$ s	completes
row-sum surrogate (one JVP)	$>40$ GB	—	out of memory
Hutchinson, $M{=}50$	$>40$ GB	—	out of memory
Hutchinson, $M{=}500$	$>40$ GB	—	out of memory

Appendix

Table 5: Memory feasibility at

128^{3}

on a

40

GB accelerator (

39.5

GiB usable), one estimator per invocation in a clean process, released implementations with activation checkpointing disabled for the tangent pass (App. K ). The three Jacobian-based estimators exhaust memory before completing a single volume; the probe differentiates nothing and completes.

Field	PNG (hybrid / head-free)	OASIS
Input resolution	$64^{3}$	$128^{3}$
Scored split	$130$ pairs, $47$ participants	$211$ pairs, $63$ participants
Forward schedule	cosine, $T=300$	cosine, $T=300$
Reverse sampler	DDIM, $\zeta=5$ , $\eta=1$	DDIM, $\zeta=5$ , $\eta=1$
MC reference	$J{=}20$ , seeds $42$ – $61$	$J{=}20$ , seeds $42$ – $61$
Self-chain probe	chain seed $42$	chain seed $42$

Appendix

Table 6: Configuration index for the medical experiments. The unscored OASIS pairs are duplicated split entries or corrupted files, excluded before any scoring. Masks and metrics are defined operationally above; the statistical protocol is below and in App. R .

checkpoint	corpus	$\rho(\nabla,e)$	exact	Hutch. $M{=}200$	row-sum	T-PT $K{=}30$
cond	coco	0.389	0.474	0.465	0.036	0.484
cond	real	0.382	0.476	0.466	0.063	0.470
cond	self	0.340	0.448	0.435	0.004	0.451
brats	self	0.297	0.320	0.320	0.015	0.315
church	self	0.130	0.224	0.224	$-0.009$	0.218
church	coco	0.064	0.165	0.165	$-0.007$	0.167

Appendix

Table 7: Spearman correlation with the denoising error, per condition, mean over

100

images. Rows are ordered by

\rho(\nabla,e)

, how much of the error is spatial structure. The block below the rule is where the exact diagonal is anti -correlated with the error. The Hutchinson column carries the same four harmful conditions as the exact column, with worst value

-0.131

and mean over the fourteen

0.128

. Every interval in both columns excludes zero except

\dagger

( ffhq/self ,

[-0.006,0.010]

for each). For T-PT , the only intervals that exclude zero on the negative side are the two marked

\ddagger

: inet/real

[-0.021,-0.007]

and inet/coco

[-0.023,-0.009]

, on the checkpoint where the diagonal reverses hardest; neither lies entirely below the

-0.02

threshold, whereas four exact-diagonal intervals do. Over the ten conditions above the rule, where the diagonal is not reversed, the exact column means

0.225

, Hutchinson

0.221

and T-PT

0.218

, and eight of the ten T-PT –exact differences lie within

0.010

; the two exceptions are inet/self and cifar/self , where the exact diagonal itself reaches only

0.058

and

0.029

. The exact column is clamped at zero wherever the implied variance is negative; the clamped fraction is zero on both pixel-space unconditional checkpoints, below

0.2\%

on the latent ones, and at most

5.7\%

on the class-conditional one.

estimator	$\bar{\rho}(\sigma,e)$	forward-equivalents
Hutchinson, $M{=}5$	$0.298$	$\approx 20$
Hutchinson, $M{=}15$	$0.359$	$\approx 60$
Hutchinson, $M{=}50$	$0.410$	$\approx 200$
Hutchinson, $M{=}200$	$0.435$	$\approx 800$
exact diagonal	$0.448$	$\approx 49{,}000$
T-PT , $K{=}30$	$0.451$	$30$

Appendix

Table 8: The

M

sweep behind Sec. 3.2 , on cond/self (the same

100

images as Table 7 ): Spearman correlation with the denoising error as the Hutchinson estimator converges to the exact diagonal. Forward-equivalents follow App. J . The pattern is the same in all fourteen conditions; at

M{=}200

Hutchinson tracks the exact column of Table 7 to within

0.013

here, and at least

0.92

in rank agreement in every one of the fourteen.

Estimator	functional of $J$	$\rho(\cdot,e)$	partial, given $J_{ii}$
Hutchinson, $M{=}50$	$J_{ii}$ , unbiased	$0.321$	$+0.056\;[+0.041,+0.070]$
Diagonal reference ( $M{=}500$ )	$J_{ii}$	$0.361$	—
Row-sum (Boys)	$\sum_{j}J_{ij}$	$0.474$	$+0.353\;[+0.283,+0.420]$
T-PT , $K{=}30$	$J_{ii}$ and off-diag. mass	$\mathbf{0.539}$	$\mathbf{+0.409}\;[+0.322,+0.489]$
Diagonal reference, given T-PT			$+0.056\;[+0.032,+0.085]$

Appendix

Table 9: Partial Spearman correlations with the error map, twenty

64^{3}

lung-CT volumes from twelve subjects, interior mask, clamped voxels excluded. The diagonal reference is Hutchinson at

M{=}500

; the one-hot exact diagonal is not affordable at this size. Intervals are a bootstrap over subjects. The last row is the reverse conditioning.

condition	exact vs. $e$	T-PT vs. $e$	T-PT , given exact	exact, given T-PT
inet/coco	$-0.132\,[-0.164,-0.101]$	$-0.016\,[-0.023,-0.009]$	$+0.005\,[+0.002,+0.008]$	$-0.131\,[-0.162,-0.100]$
inet/real	$-0.131\,[-0.165,-0.098]$	$-0.014\,[-0.021,-0.007]$	$+0.005\,[+0.001,+0.008]$	$-0.130\,[-0.164,-0.097]$
cifar/coco	$-0.091\,[-0.117,-0.064]$	$+0.004\,[-0.002,+0.011]$	$+0.007\,[+0.001,+0.013]$	$-0.091\,[-0.117,-0.065]$
cifar/real	$-0.077\,[-0.105,-0.049]$	$+0.018\,[+0.012,+0.024]$	$+0.020\,[+0.014,+0.025]$	$-0.077\,[-0.106,-0.049]$

Appendix

Table 10: The same partial analysis on the four conditions where the exact diagonal is anti-correlated with the error,

100

images each, full image, intervals, a bootstrap over images. Here the conditioning variable is the one-hot exact diagonal itself, not a Hutchinson reference, and no voxel clamps. The first two columns reproduce Table 7 . Removing the probe from the diagonal leaves the diagonal’s reversal where it was; removing the diagonal from the probe leaves a small positive partial that excludes zero in every condition. These partials describe the finite-budget maps and do not, by themselves, establish complementary information in the population estimand. They are also not comparable in reliability with the clinical partials of Table 9 : the attenuation bound

|\rho|\leq\sqrt{\mathrm{rel}}

places the clinical map’s reliability at

\geq 0.45

and these four at

\leq 0.02

, so the contrast between the two tables is a probe budget and not two verdicts on the functional.

condition	$a\overline{\\|J_{i:}\\|^{2}}/\overline{\|J_{ii}\|}$	$\rho(\\|J_{i:}\\|^{2},e)$	$\rho(\sigma_{\text{T2}},e)$	$\rho(\sigma_{\text{TW}},\sigma_{\text{T2}})$	T-PT , given $\sigma_{\text{T2}}$
inet/coco	$0.34$	$+0.135\,[+0.104,+0.167]$	$-0.131\,[-0.163,-0.099]$	$0.141$	$+0.005\,[+0.002,+0.007]$
inet/real	$0.34$	$+0.134\,[+0.100,+0.167]$	$-0.130\,[-0.163,-0.096]$	$0.143$	$+0.005\,[+0.001,+0.008]$
cifar/coco	$0.28$	$+0.093\,[+0.066,+0.119]$	$-0.091\,[-0.117,-0.064]$	$0.023$	$+0.007\,[+0.001,+0.013]$
cifar/real	$0.28$	$+0.079\,[+0.051,+0.106]$	$-0.077\,[-0.105,-0.050]$	$0.013$	$+0.020\,[+0.014,+0.026]$
cond/self	$0.98$	$-0.442\,[-0.471,-0.413]$	$+0.431\,[+0.402,+0.460]$	$0.840$	$+0.174\,[+0.159,+0.189]$

Appendix

Table 11: The row energy, read directly. The same exact-Jacobian pass accumulates

\|J_{i:}\|^{2}

alongside

J_{ii}

on the four reversed conditions and, for contrast, on the self-conditioned ImageNet model ( cond/self , the best-behaved condition of Table 7 );

100

images each, full image, intervals a bootstrap over images.

\sigma_{\text{T2}}=(1-2aJ_{ii}+a^{2}\|J_{i:}\|^{2})^{1/2}

is Eq. ( 4 ) evaluated exactly, with

a=\sigma_{t^{\ast}}

. The first column is the quadratic term against

\overline{|J_{ii}|}

; the linear term of Eq. ( 4 ) carries a further factor of two, so the quadratic term matches it at a column value of

2

. On the reversed conditions the row energy is the mirror image of the diagonal (

\rho(\|J_{i:}\|^{2},\sigma_{\text{exact}})

between

-0.984

and

-0.999

), because

\|J_{i:}\|^{2}

is dominated there by

J_{ii}^{2}

; the second-order map ranks identically to the exact diagonal (

\rho=1.000

on all four,

0.952

on cond/self ) and reverses with it; and the probe retains conditioned on

\sigma_{\text{T2}}

exactly what it retains conditioned on

J_{ii}

(Table 10 ). The

K{=}30

map correlates with that combination at only

0.01

0.14

on the reversed conditions, and Table 12 says why: at that budget the map is barely reproducible there. The low agreement is consistent with this poor reproducibility and cannot, by itself, establish a difference in population rankings. On cond/self , where the map is reproducible, it agrees at

0.840

and still retains

+0.174

beyond it.

	$\sigma_{\text{T2}}$	T-PT , $\rho(\sigma_{K},e)$ [95% CI]			Hutch. $M{=}5$
condition	vs. $e$	$K{=}30$	$K{=}120$	$K{=}480$	[95% CI]
cifar/coco	$-0.091$	$+0.004\,[-0.002,+0.010]$	$-0.014\,[-0.020,-0.008]$	$-0.007\,[-0.016,+0.001]$	$-0.043\,[-0.057,-0.029]$
cifar/real	$-0.077$	$+0.018\,[+0.012,+0.024]$	$-0.015\,[-0.021,-0.009]$	$-0.009\,[-0.017,-0.001]$	$-0.042\,[-0.057,-0.028]$
inet/coco	$-0.131$	$-0.016\,[-0.023,-0.010]$	$-0.041\,[-0.052,-0.030]$	$-0.079\,[-0.099,-0.059]$	$-0.104\,[-0.131,-0.077]$
inet/real	$-0.130$	$-0.014\,[-0.021,-0.007]$	$-0.031\,[-0.043,-0.020]$	$-0.067\,[-0.087,-0.047]$	$-0.101\,[-0.129,-0.074]$
cond/self	$+0.431$	$+0.451\,[+0.422,+0.479]$	$+0.465\,[+0.436,+0.493]$	$+0.471\,[+0.443,+0.499]$	$+0.298\,[+0.272,+0.326]$

Appendix

Table 12: Probe budget on the reversed conditions, with cond/self as the informative control. Upper block: correlation with the error at three budgets from one run, each with its own bootstrap interval over the

100

images (

B=2\times 10^{4}

); a bracket belongs to the number on its left. The

K=30

column reproduces the exact-Jacobian pass to the last digit. Probe sets at different

K

are drawn independently. Lower block: reliability of a

g

-probe map, the Spearman agreement between the maps of two disjoint groups of

g

probes from the

K=480

draws, so

g=30

is the reproducibility of the map the body reports. The last column is Hutchinson at

M=5

, about

20

forward-equivalents, level above and split-half reliability below. Two facts carry the appendix: at matched budget Hutchinson is harmful on all four and the probe on none, and given

4

–

16\times

the probes the probe’s inet maps become more reproducible and harmful with them.

Figure 3: Representative lung-CT sample ( left ) and OASIS-3 transverse slice ( right ). Top row of each: baseline, target, absolute error. Bottom row, overlaid on the target: MC-DDIM $\sigma$ , the analytical $\sigma_{\theta}$ , and T-PT $\sigma_{\text{TW}}$ . Each overlay is normalised to its own quantiles, so the panels show each map’s spatial ordering and not comparable magnitudes. On lung CT all three concentrate on the nodule region and T-PT reproduces MC-DDIM’s spatial pattern (cross-agreement $\rho=0.79$ on this volume) at $13.4\times$ fewer network evaluations. On brain MRI $\sigma_{\text{TW}}$ tracks the ventricular and cortical boundaries consistently with MC-DDIM ( $\rho=0.70$ here, $0.709$ over the cohort) and inside brain tissue it leads the ensemble on five of the eight endpoints of Table 1 , the rest unresolved. What no panel shows, and Sec. 6 reports, is that both maps lose most of their information inside the region that actually changed between the two visits.

probe input $x_{0}$ (lung CT, $N{=}130$ )	source	$\bar{\rho}(\sigma_{\text{TW}},e)$
$[x_{\text{baseline}},0]$ , zero placeholder in the residual channel	first implementation	$0.004$
one DDIM reconstruction (deployment probe)	Table 14	$0.648$

Appendix

Table 13: The probe point is load-bearing (Sec. 4.3 ): the same estimator, at the same

t^{\ast}=60

and

K=30

, scored against the error of the chain each map accompanies — the deployment scoring defined above, not the common error map — with only the probe input changed. Both rows are per-volume means, so the two are read against each other; the participant-weighted figure for the deployment probe is the

0.613\;[0.576,0.650]

of Table 14 . A placeholder that no training pair ever visited yields a smooth, non-negative map that carries no information about the error; any model-supported reconstruction restores it.

Dataset	Estimator	Fwd.	$\bar{\rho}(\sigma,e)$ [95% CI]	cross $\leftrightarrow$ MC	Requires
PNG	$\sigma_{\text{MC}}$ ( $J{=}20$ )	$1220$	$0.642\;[0.596,0.687]$	$\equiv 1$	—
$N{=}130$	$\sigma_{\theta}$ (analytical, $\hat{s}{=}15$ )	$61$	$0.597\;[0.551,0.641]$	—	hybrid loss
	$\sigma_{\text{TW}}$ ( T-PT )	$\mathbf{91}$	$0.613\;[0.576,0.650]$	$0.790\;[0.765,0.812]$	—
OASIS	$\sigma_{\text{MC}}$ ( $J{=}20$ )	$1220$	$\mathbf{0.552}\;[0.538,0.564]$	$\equiv 1$	—
$N{=}211$	$\sigma_{\theta}$ (analytical, $\hat{s}{=}15$ )	$61$	$\mathbf{0.551}\;[0.539,0.562]$	—	hybrid loss
	$\sigma_{\text{TW}}$ ( T-PT )	$\mathbf{76}$	$0.520\;[0.511,0.529]$	$0.709\;[0.705,0.712]$	—

Appendix

Table 14: Predictive discrimination in the deployment setting: each estimator is scored against the error of the prediction it would actually accompany, so T-PT is measured against a single chain and MC-DDIM against its

J

-chain mean. “Fwd.” is network evaluations per volume; CI is confidence interval.

\sigma_{\theta}

accumulates along its own single chain, so its

61

are that chain itself, with no evaluations beyond it.

\sigma_{\text{MC}}

is the reference, so its cross-agreement with itself is

1

by construction; entries are participant-weighted with participant-level bootstraps on both tasks. The two

\sigma_{\theta}

rows are scored against the common error map (their cross-agreement with MC-DDIM was not computed); on lung CT the analytical baseline sits below both estimators, and T-PT needs no variance head to run, at

13.4\times

fewer network evaluations than the reference. On brain MRI, over the whole volume, MC-DDIM is ahead of T-PT by

3.2

Spearman points in this self-scored setting (about four against the common error map); Table 1 shows the ordering reversing inside tissue, which is where the background stops carrying the correlation. The tissue rows of both analytical baselines are in Table 17 .

Lung CT (PNG, $N{=}130$ , $G{=}47$ )
Metric	participant-wtd. $\bar{\Delta}$ [CI]	volume-wtd. $\bar{\Delta}$ [CI]	T-PT wins
$\rho$ , full volume	$-0.0215\;[-0.0373,-0.0070]$	$-0.0242\;[-0.0376,-0.0104]$	$36/130$
$\rho$ , tissue	$-0.0215\;[-0.0368,-0.0069]$	$-0.0242\;[-0.0375,-0.0103]$	$36/130$
$\rho$ , edges	$-0.1465\;[-0.1865,-0.1096]$	$-0.1760\;[-0.2102,-0.1295]$	$17/130$
$\rho$ , interior	$-0.0234\;[-0.0435,-0.0048]$	$-0.0210\;[-0.0413,-0.0062]$	$50/130$
$\rho$ , regions of change	$-0.1337\;[-0.1821,-0.0860]$	$-0.1656\;[-0.1945,-0.1171]$	$30/130$

Appendix

Table 15: Paired participant-level bootstrap of

\sigma_{\text{TW}}-\sigma_{\text{MC}}

against the common error map, on both tasks: the

130

lung-CT pairs from

47

participants and all

211

OASIS volumes from the

63

participants of the published test split.

\Delta

is T-PT minus MC-DDIM on every row, so on AUSE and AURC, a positive value favours MC-DDIM. The participant-weighted column averages the difference within a participant before averaging across participants, so a participant contributing

21

of the

211

pairs counts once rather than

21

times; the volume-weighted column averages over volumes directly. Both bootstraps resample participants. The two agree in sign on every row of both tasks. The last column counts the volumes on which T-PT is better, with the direction of each metric taken into account. Tissue Spearman is the pre-designated primary endpoint and the gradient-controlled row its robustness check (App. M ); the remaining rows are secondary, unadjusted for multiplicity, and any one of them alone should be read as exploratory. CI is the confidence interval.

	PNG ( $N{=}130$ )		OASIS ( $N{=}211$ )
	$\sigma_{\text{TW}}$	$\sigma_{\text{MC}}$	$\sigma_{\text{TW}}$	$\sigma_{\text{MC}}$
$\rho$ , full volume	$0.669$	$\mathbf{0.693}$	$0.518$	$\mathbf{0.559}$
$\rho$ , tissue	$0.670$	$\mathbf{0.694}$	$\mathbf{0.349}$	$0.337$
$\rho$ , edges	$0.352$	$\mathbf{0.528}$	$0.248$	$0.245$
$\rho$ , interior	$0.612$	$\mathbf{0.633}$	$\mathbf{0.334}$	$0.318$
$\rho$ , regions of true change	$0.100$	$\mathbf{0.265}$	$0.128$	$0.133$

Appendix

Table 16: Full per-region and selective-prediction results against the common error map of the

J=20

ensemble mean, expanding the compact Table 1 of the body. OASIS entries are per-volume means, shown for scale, following the convention of Table 1 ; the pre-declared participant-weighted differences and their intervals for OASIS are in Table 15 ; the PNG paired contrasts are the

\Delta

column of Table 1 . Bold marks the better of

\sigma_{\text{TW}}

and

\sigma_{\text{MC}}

where the

95\%

paired interval excludes zero (participant-level resampling on both tasks); intervals are in Table 15 . “Tissue” is the foreground mask. On lung CT it covers

99.5\%

of the volume; the scan is already cropped to the thorax, so foreground and full volume coincide, and the row is a consistency check rather than a result. On brain MRI it covers

24\%

, and there the two rows differ sharply. The

\sigma_{\theta}

columns for both tasks are in Table 17 .

Lung CT (PNG, $N{=}130$ , $G{=}47$ )
	$\sigma_{\theta}$ (analyt.)	$\sigma_{\text{TW}}$	$\sigma_{\text{MC}}$
$\rho$ , full volume	$0.643$	$0.669$	$\mathbf{0.693}$
$\rho$ , tissue	$0.644$	$0.670$	$\mathbf{0.694}$
$\rho$ , partial, $\|\nabla u_{0}\|$ ctl.	$0.295$	$0.339$	$\mathbf{0.409}$
Network evaluations / volume	$61$	$91$	$1220$
Paired contrasts against $\sigma_{\theta}$ (participant-weighted, $B=2\times 10^{4}$ ):

Appendix

Table 17: The analytical variance-propagation baseline on both tasks, from one seed-

42

chain per volume and scored against the common

J{=}20

ensemble-mean error map. Estimator rows are per-volume means, following the convention of Table 1 ; contrast rows are the participant-weighted paired difference (column estimator minus

\sigma_{\theta}

), with brackets giving 95% confidence intervals (CI), where positive favours the column estimator except for AUSE, where negative does. AUROC is reported for the worst-

5\%

-error voxels. On lung CT, both estimators beat the variance head on every endpoint, the reference by more. On brain MRI, inside tissue both estimators beat it and T-PT by more; over the whole volume neither does, the T-PT contrast being negative and the MC one crossing zero. In the OASIS regions of change (not shown) no contrast against

\sigma_{\theta}

excludes zero in either direction.

		probe seed $42$	alternative probe seed
OASIS	$\Delta\rho$ , tissue	$+0.0165\;[+0.0056,+0.0282]$	$+0.0149\;[+0.0038,+0.0268]$
	$\Delta\rho$ , partial (grad. rm.)	$+0.0212\;[+0.0088,+0.0348]$	$+0.0193\;[+0.0068,+0.0332]$
	$\Delta\rho$ , full volume	$-0.0380\;[-0.0415,-0.0346]$	$-0.0374\;[-0.0408,-0.0339]$
	$\rho(\sigma_{\text{TW}},\sigma_{\text{MC}})$	$0.709$	$0.709$
PNG	$\Delta\rho$ , tissue	$-0.0215\;[-0.0372,-0.0069]$	$-0.0256\;[-0.0428,-0.0093]$
	$\Delta\rho$ , partial (grad. rm.)	$-0.0595\;[-0.0896,-0.0306]$	$-0.0650\;[-0.0960,-0.0353]$

Appendix

Table 18: Probe-seed sensitivity: the masked pass run with the deployment probe seed (

42

) and with a probe seed outside the reference seed set (

42

–

61

): seed

777

on OASIS, seed

123

on lung CT, everything else held fixed, on the deployment conditioning and against the same references as Table 1 .

\Delta\rho

is the paired difference, T-PT minus MC-DDIM, of the per-volume Spearman correlation with the common error map; the last row of each block is the Spearman correlation between the two

\sigma

maps themselves. The table is read for the seed-to-seed movement only; contrasts are participant-weighted with participant-level intervals (

B=10^{4}

) on both tasks. Every primary contrast moves by at most

0.006

Checkpoint	Contrast $\Delta$	$\bar{\Delta}$ [95% CI]	$P(\Delta>0)$
PNG hybrid	$\sigma_{\text{TW}}-\sigma_{\text{MC}}$	$-0.022\;[-0.037,-0.007]$	$<10^{-3}$
PNG no-head	$\sigma_{\text{TW}}-\sigma_{\text{MC}}$	$-0.020\;[-0.040,-0.001]$	$0.02$

Appendix

Table 19: Paired bootstrap on the two lung-CT checkpoints against the error map of each checkpoint’s own

J{=}20

ensemble, resampling the

47

participants (

B=2\times 10^{4}

); the conditioning of each experiment is given in App. M and App. L . MC-DDIM is ahead of T-PT by about two Spearman points on both checkpoints; on the one trained without a variance head, the interval only just excludes zero. The sample-level bootstrap used in an earlier draft gave

-0.003\;[-0.016,+0.009]

for that checkpoint, which is the coverage failure App. R documents. The OASIS contrasts are in Table 15 .

$J^{\prime}$	$\bar{\rho}(\sigma_{J^{\prime}},e)$	split-half	$\bar{\rho}(\sigma_{J^{\prime}},e)$	agree w/ $J{=}20$	split-half	$\bar{\rho}_{\text{tissue}}(\sigma_{J^{\prime}},e)$
	PNG ( $N{=}130$ )		OASIS ( $N{=}211$ )
$2$	$0.529$ ( $77\%$ )	—	$0.485$ ( $91\%$ )	$0.668$	—	$0.139$ ( $43\%$ )
$3$	—	—	$0.521$ ( $98\%$ )	$0.757$	—	$0.195$ ( $60\%$ )
$5$	$0.653$ ( $95\%$ )	$0.496$	$0.529$ ( $99\%$ )	$0.827$	$0.535$	$0.248$ ( $77\%$ )
$8$	—	—	$0.531$ ( $100\%$ )	$0.887$	$0.665$	$0.284$ ( $88\%$ )
$10$	$0.676$ ( $98\%$ )	$0.768$	$0.532$ ( $100\%$ )	$0.915$	$0.684$	$0.296$ ( $91\%$ )

Appendix

Table 20: Percentages are relative to

J^{\prime}=20

. Each entry averages

20

subsets of size

J^{\prime}

drawn at random without replacement, so the curve does not depend on the order of the seeds. Entries are volume-weighted means computed by the standalone convergence probe directly in prediction space, so its absolute values are not comparable with the scored pipeline of Tables 1 – 16 ; only the percentages, which are internal to this table, are meaningful here. The last column is the finding: full-volume discrimination saturates almost immediately (

J^{\prime}=3

recovers

98\%

), but that saturation is carried by background. Inside tissue , the primary endpoint, the same reference recovers only

60\%

J^{\prime}=3

91\%

J^{\prime}=10

, and is still climbing at

J^{\prime}=20

. Among the tested configurations

J^{\prime}\leq 20

, no smaller ensemble reaches the

J{=}20

tissue discrimination, which is itself still increasing.

		$J{=}20$		$J{=}50$
		T-PT	MC	T-PT	MC
Lung CT	$\rho$ , tissue	$0.670$	$0.694$	$0.671$	$0.702$
$N{=}130$ , $G{=}47$	$\Delta\rho$	$-0.022\,[-0.037,-0.007]$		$-0.025\,[-0.039,-0.012]$
	$\Delta\rho$ , interior	$-0.023\,[-0.043,-0.005]$		$-0.026\,[-0.044,-0.009]$
	$\Delta\rho$ , partial (grad. rm.)	$-0.060\,[-0.090,-0.031]$		$-0.057\,[-0.087,-0.029]$
	$\Delta$ AUROC	$-0.027\,[-0.039,-0.017]$		$-0.033\,[-0.044,-0.022]$

Appendix

Table 21: Rescoring against a

J{=}50

reference.

\Delta\rho

and

\Delta

AUROC are T-PT minus MC-DDIM, as everywhere else; the lung-CT

J{=}20

intervals are those of Table 1 . The lung-CT rows are the full cohort with participant-level intervals; the brain rows are the first thirty volumes, which come from five participants, so they are reported as a convergence check and carry no intervals — resampling five clusters would not mean anything. The brain

J{=}20

column is recomputed on those same thirty volumes, so the two columns are paired; it is larger there (

+0.038

) than over the full cohort (

+0.016

, Table 1 ), which is why only the within-table comparison is read. At fifty chains the ensemble costs

3{,}050

network evaluations per volume against the probe’s

76

, and inside brain tissue the two are then close on every endpoint, AUROC included. The last row counts volumes on which T-PT has the higher tissue

\rho

	$t^{\ast}{=}15$	$t^{\ast}{=}30$	$t^{\ast}{=}60$	$t^{\ast}{=}120$	MC-DDIM
$\rho$ , tissue	$0.663$	$0.671$	$0.670$	$0.669$	$0.694$
$\rho$ , interior	$0.619$	$0.621$	$0.612$	$0.606$	$0.633$
$\rho$ , edges	$0.244$	$0.296$	$0.352$	$0.406$	$0.528$
$\rho$ , change region	$0.042$	$0.065$	$0.100$	$0.145$	$0.265$
$\rho$ , partial (grad. rm.)	$0.317$	$0.334$	$0.339$	$0.346$	$0.409$
AUROC, top-5% error	$0.854$	$0.870$	$0.884$	$0.897$	$0.922$

Appendix

Table 22: Sensitivity to the probe timestep on lung CT (

N{=}130

G{=}47

), self-chain probe,

K=30

, scored against the common error map; absolutes are per-volume means, the paired contrasts are participant-weighted, and every contrast interval excludes zero. Tissue

\rho

moves by less than

0.01

between

t^{\ast}=15

and

120

, the gradient-controlled partial by

0.03

. The regions where the probe trails MC-DDIM, edges and true change, improve monotonically with

t^{\ast}

and at

120

remain well below it.

t^{\ast}=60

was fixed before any of these runs and is kept.

	PNG ( $N{=}130$ )
$K$	Fwd.	$\bar{\rho}(\sigma,e)$	% of $K{=}30$
$5$	$66$	$0.567\;[0.530,0.603]$	$92.4\%$
$10$	$71$	$0.595\;[0.557,0.631]$	$96.9\%$
$15$	$76$	$0.604\;[0.566,0.639]$	$98.4\%$
$30$	$91$	$0.613\;[0.575,0.649]$	$100\%$

Appendix

Table 23: Probe size, self-chain probe,

t^{\ast}=60

, lung CT, participant-weighted with participant-level intervals; percentages relative to

K=30

. Returns flatten quickly:

K=10

recovers

96.9\%

and

K=15

recovers

98.4\%

. The reverse chain dominates the per-volume cost, so the sweep is obtained from the same chain and covers the whole test set rather than a subset. We report

K=30

on PNG and

K=15

on OASIS, both fixed before any 63-participant score was read; nothing in the method depends on those choices. The maps are scored against the self-chain (deployment) error map, as in Table 14 .

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