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One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion

Gordei Verbii

arXiv:2608.23094v1stat.MLcs.LG

TL;DR

The paper asks whether one implicit DDIM inversion step genuinely measures local manifold geometry and derives its Bayes-limit calibration and validity conditions. It formulates inversion as a strongly convex potential problem, analyzes contraction and curvature-controlled convergence, and tests exact and trained scores. The results separate guaranteed solver and model certificates from geometry readouts that remain limited by training support and validity windows.

  • Problem

    One implicit DDIM inversion step is widely used as a cheap probe of whether pretrained diffusion models encode local manifold geometry, but what it measures and when its readouts are valid require calibration.

  • Method

    The paper derives a Bayes-limit potential, posterior-covariance bounds, contraction constants, curvature-domain laws, and calibration identities, then compares exact Bayes denoisers with trained scores.

  • Results

    The inversion potential has modulus e^-h_t, contraction constant ρ_g^star = 1-e^-h_t < 0.326 on the standard DDPM schedule, and exact scores reproduce the predicted shells while trained scores violate the unconditional ceiling.

  • Takeaways & Limitations

    Uniqueness, contraction, and score-ceiling violations can be interpreted without geometric hypotheses, whereas geometry readouts require a calibrated convergence window.

  • Takeaways & Limitations

    No trained score shows a convergence shell because the reachable Fermi window conflicts with training support, while finite-noise curvature laws remain expansion results within that window.

Abstract

from arXiv · show

One implicit DDIM inversion step is the cheapest probe of whether a pretrained diffusion model encodes local manifold geometry. It is the stationarity condition of an explicit potential, $x-G(x)=\nablaΨ_t(x)$, strongly convex at the Bayes limit with modulus exactly $e^{-h_t}$ for the step's log-SNR gap $h_t$ $-$ for every data law, schedule and point, with no manifold, reach or unimodality hypothesis. Three consequences must be kept apart. (i) The solution is unique at the Bayes limit; a second one requires the trained score to violate the posterior-covariance bound by $1/(1-e^{-h_t})$, a hypothesis-free certificate of model error; the same bound makes contraction a schedule constant, $ρ_g^{\star}=1-e^{-h_t}<0.326$ throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unit-step gradient descent on $Ψ_t$, unstable wherever $λ_{\max}(\nabla^2Ψ_t)>2$, so oscillation certifies nothing; damping below $2/λ_{\max}$ cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth $w=rκ_{\max}$ the oscillation shell sits at $w=\tfrac12$, schedule-free, and the divergence shell at $w=1/(1+ρ_g^{\star})$, with a measured finite-noise correction in $\|\mathrm{II}\|^2$. Exact scores reproduce both to within $0.54\%$ on three classes; no trained score we probe shows a shell $-$ a derived limitation, not a null result: the Fermi window conflicts with the model's own training support by $3.6$-$5.6\times$, and the trained Hessian-Lipschitz constant is $2$-$12\%$ of the curvature the law reads, $0$ on a ReLU net. Finally the unconditional ceiling $σ_tλ_{\max}(\mathrm{sym}\,J)\le1$, from $\mathrm{Cov}(x_0\mid x_t)\succeq0$ alone, holds for the exact score to $3\times10^{-7}$ but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by $1.26$-$4.66\times$.

1 INTRODUCTION

The paper reframes one implicit DDIM inversion step as a calibrated convex-program probe of diffusion-model geometry. At Bayes optimality, uniqueness and contraction are schedule-controlled, while solver instability and convergence-domain geometry are distinct phenomena.

  • Probe and calibration: One implicit DDIM inversion step is the stationarity condition of an explicit potential and provides a calibration curve for local manifold-geometry readouts.The readouts are meaningful only within a noise-scale window bounded by reach and the model’s calibration.
  • Solver versus geometry: Picard instability does not contradict uniqueness: its unit step can oscillate, while the convergence domain is governed by curvature shells at schedule-fixed fractions of focal radius.The oscillation shell is w = 1/2 and the divergence shell is w = 1/(1+ρ_g^star).
  • Convex inversion: The inversion potential is strongly convex at Bayes optimality, with modulus e^-h_t, making the solution unique for every data law and schedule.The same result separates uniqueness from solver behavior and model geometry.
  • Contraction and multiplicity: The Bayes contraction constant is ρ_g^star = 1-e^-h_t and remains below 0.326 throughout the standard DDPM schedule.A second solution therefore certifies violation of the posterior-covariance bound by the trained score.
  • Empirical scope: Exact scores reproduce the predicted calibration and convergence boundaries, whereas trained scores show no shell because the reachable Fermi window conflicts with training support.The paper presents this as a derived limitation rather than a null result.

2 RELATED WORK

The paper situates its contribution among work reading manifold geometry from diffusion scores, analyzing fixed-point solvers, and deriving score-based geometric quantities. It claims novelty for the inversion potential, schedule constant, convergence-domain law, and related calibrations rather than for the antecedent identities themselves.

  • Score geometry: Prior work uses low-noise score direction, score rank, Fokker–Planck functionals, conservativity, DSM loss, and spectral phases to study local geometry.These strands motivate reading intrinsic structure from pretrained diffusion models.
  • Inversion solvers: Existing inversion methods solve related residuals with Picard, Anderson, Newton–Krylov, or regularized variants, but the paper distinguishes their solver targets.Picard, Anderson, and Newton–Krylov agree closely on one residual, while regularized and averaged variants can shift the root.
  • Convergence theory: The analysis builds on Kantorovich and spectral-radius results, using a closed-form Bayes covariance structure instead of an unavailable Smale γ constant.This supplies a direct route to convergence analysis at Bayes optimality.
  • Positioning: The paper claims no novelty for the normal-block constant, Tweedie–Miyasawa identity, normalized spectrum, or codimension-scaled window.Its stated novelties are the potential and modulus, schedule contraction constant, convergence-domain law, covariance ceiling, and additional identities.

3 PRELIMINARIES AND PROBLEM SETUP

The setup models data near a low-dimensional manifold, defines the diffusion score and denoiser Jacobians, and parameterizes one DDIM inversion step by its log-SNR gap. The resulting probe evaluates a trained noise predictor at the unknown noisier latent.

  • Data and manifold: Data lie near a d-dimensional submanifold of R^D, while diffusion models operate on noise-convolved data and estimate its score.The manifold and reach assumptions support the geometric analysis where explicitly invoked.
  • Objects and evaluation points: The notation distinguishes the sampled point x_t from the noiseless point x° = √ᾱ_t x_0, and defines the trained and Bayes-optimal denoisers with their Jacobians.This distinction is central to interpreting probe evaluations.
  • Notation: The setup defines denoiser-loss quantities L_t, Π_t, and δ_t alongside singular values, curvature, sample budget, and probe-count notation.These quantities support later calibration and estimator identities.
  • Schedule and inversion: The inversion step is parameterized by λ_t, the log-SNR gap h_t, and schedule coefficients A_t and B_t, with A_tσ_t = e^-h_tσ_t+1.Schedule constants therefore depend on the step’s log-SNR gap.
  • Fixed-point problem: Inverting from target y = x_t solves for x = x_t+1 using iterative methods, with the network evaluated at the unknown noisier latent at index t+1.Substitution of a trained predictor produces a map whose Lipschitz factor can be unbounded in the stride.

4 PROPOSED METHOD

The method derives Bayes-limit identities for score Jacobians and turns implicit inversion into a convex program with explicit uniqueness, contraction, and curvature-domain results. It also identifies which readouts are unconditional and which require geometric validity windows.

  • Normal-bundle law: At Bayes optimality, the score Jacobian is governed by posterior covariance, yielding a normal-bundle stiffness ceiling and curvature-dependent singular-value behavior.The ceiling is hypothesis-free, while curved manifolds can make the largest singular value strictly below or above 1/σ_t.
  • Convex program: The residual is a gradient field whose potential has Hessian at least e^-h_t I, so Ψ_t is strongly convex and Picard is unit-step gradient descent.The condition number and distance to ill-posedness are schedule-controlled at Bayes optimality.
  • Uniqueness and multiplicity: Bayes optimality guarantees a unique fixed point, while a second solution requires a trained-score covariance violation and is repelling for Picard.Multiplicity is observable through Newton-type behavior rather than failed Picard iteration.
  • Contraction: Under the stated assumptions, the inversion map has schedule-only contraction constant ρ_g^star = 1-e^-h_t, and damping with η < 2/Λ_t stabilizes Picard.Newton and Anderson are insensitive to the local Hessian upper bound Λ_t.
  • Convergence domain: Along principal normals, exact Picard multipliers place the oscillation boundary at w_osc = 1/2 and divergence boundary at w_div = 1/(1+ρ_g^star), with finite-noise corrections from the second fundamental form.The boundaries are fractions of focal radius, while the Kantorovich product diverges as w approaches 1.
  • Further readouts: The method supplies further Bayes values for Hamilton–Jacobi, scaling, loss, rank, and sampled-point dimension readouts, distinguishing forced evaluations from geometry-sensitive ones.At sampled points, the Fokker–Planck dimension functional has an expectation fixed by model orthogonality defect rather than manifold dimension.

5 EXPERIMENTS

Experiments validate the Bayes-limit calibration on exact scores, expose failures of trained-model geometry readouts, and distinguish model error from solver instability and estimator limits.

  • Protocol: The protocol probes pretrained DDPMs, synthetic conditional score models, and exact Bayes denoisers through an identical harness.The exact-score arm isolates estimator failure from model failure.
  • Stiffness and contraction: All trained scores violate the unconditional stiffness ceiling, while the exact score reaches it within 3×10^-7.Violations span cloud, CIFAR-10, and CelebA-HQ-256 settings, with the largest image excess reaching 4.6637×.
  • Stiffness and contraction: The schedule-only contraction prediction matches the exact score but trained networks show an 18–42× deficit and can diverge under a larger stride.At the CIFAR probe, the measured Picard rate is 0.1409–0.1498 versus ρ⋆_g=0.1180, while the Bayes map contracts at 0.7134 under the divergent step.
  • Geometry readouts: Machine-precision oracle controls recover flattened support dimensions and distinguish d=1 from d=2, whereas trained models fail these geometry readouts.The exact-score harness recovers targets across five classes, while trained models return inflated or nondiscriminating values.
  • Convergence domain: The exact score reproduces both convergence shells near the predicted locations, including the finite-noise curvature correction.Across the sphere, two spheres, and torus, the oscillation shell is near w=1/2 and the divergence shell near 1/(1+ρ⋆_g), with a measured correction confirmed on the torus.
  • Convergence domain: 3.6× shortfall leaves no jointly valid Fermi window and convergence-shell setting in the tested protocol.The reachable depth is capped below the oscillation shell, so the absence of trained shells is derived from the protocol and model support rather than observed absence.
  • Field-based calibration: The exponent readout cannot discriminate geometry when transverse spectra are degenerate, returning θ=0.5000 with R^2=1.000000 on the Morse–Bott energy.A genuinely quartic energy instead gives 0.077, while the field estimator returns the true exponent on all four tested cases.
  • Field-based calibration: The field-based exponent readout recovers θ=0.50000 on exact scores and gives θ=0.468–0.523 on two of three CelebA-HQ-256 images, unlike descent estimates below 0.01.The CelebA fits have R^2≥0.9998, but the intercept fails everywhere, so only the slope is reported.

6 DISCUSSION AND CONCLUSION

The discussion separates solver behavior, geometric observability, and model limitations. It concludes that several image readouts are hypothesis-free, while reach- and tube-dependent claims remain uncheckable on natural images.

  • Image-scope limitations: On natural images, exactly four reported readouts are hypothesis-free, while every reading requiring reach, a Fermi window, or a tube edge is uncheckable.The four hypothesis-free readouts are the spectral ceiling, skew index, trace bound, and sampled-point identity.
  • Convergence-domain observability: The convergence shell is structurally unmeasurable because the Fermi window and training support limit observable depth to w ≤ 0.14.This falls 3.6× short of the oscillation shell and 5.6× short of the divergence shell, with 0 of 25 trained settings satisfying both conditions.
  • Unresolved measurements: The convergence-domain law remains open on trained scores because the measured Hessian–Lipschitz constant is only 2–12% of the curvature the law reads.The estimate is a two-point σ-fit on one network with no error bar.
  • Scope boundaries: The focal-radius law is claimed for the Bayes limit and oracle; finite-noise corrections and radius-to-curvature conversion require the Fermi window and roughly constant κmax.The multiplier statement alone holds at every σt and for every data law.
  • Scope boundaries: The paper’s two training states and two networks are snapshots from one recipe, so trained class-to-class comparisons have no population error bar.This limits how broadly the observed trained-score behavior can be generalized.
  • Convergence-domain observability: No trained score produces either convergence-domain shell, but this is an instrument-resolution result rather than evidence that trained scores lack such structure.The exact score reproduces both closed-form boundaries, whereas trained-score measurements are blocked by the probe’s observable range.

A THE CALIBRATION SUITE

The calibration suite evaluates exact and schedule-dependent quantities underlying one-step inversion. It establishes a schedule-only contraction constant, clarifies schedule edge cases, and quantifies parametrization invariance.

  • Exact-score suite: The Gaussian convolution is available in closed form for the calibration suite’s final case.
  • Exact-score suite: The exact-score calibration differentiates ε⋆ = −σt∇log pt and J⋆ = −σt∇²log pt to machine precision.For the Bessel-ratio case, the implementation uses exponentially scaled evaluation and the Riccati identity.
  • Contraction calibration: ρg⋆ = 1 − e^-ht is the corrected contraction constant when the score is evaluated at t+1.Normalizing by σt instead inflates the constant by exactly eht; under the standard DDPM linear schedule its maximum is 0.3256994484136135.
  • Schedule dependence: A cosine schedule with ᾱT = 0 reaches ρg⋆ → 1 at its last step, so printed margins reflect library floors or clipping rather than the nominal schedule itself.The diffusers convention gives 0.968377, whereas clipping ᾱ at 10^-8 gives 0.935834.
  • Schedule dependence: The bound satisfies ρg⋆ < 1 for schedules with ᾱT > 0, while equality occurs only at a zero-terminal-SNR top step.Rectified flow attains ρg⋆ = 1.000000 at its top step; cosine-schedule margins depend on clipping conventions when ᾱT = 0.
  • Schedule dependence: EDM/Karras spans ρg⋆ values from 0.024484 at stride 1 to 0.681146 at stride 50, so the certificate margin varies by a factor of 28 across families.The contraction bound remains schedule-defined, but its numerical margin depends strongly on the schedule family and stride.
  • Parametrization invariance: Log-SNR-axis translations leave ρg⋆ unchanged, with max |hVE − hEDM| = 2.9 × 10^-16 over 400 matched σ-grids.Static and dynamic shifts likewise change ρg⋆ by at most 3.8 × 10^-14.

B FULL PROOFS

The proofs recast one implicit inversion step as a schedule-controlled strongly convex problem, separating root uniqueness, solver stability, and geometric information in the convergence domain.

  • Potential and convexity: The exact residual is the gradient of an explicit potential, whose Hessian satisfies ∇^2Ψ_t ⪰ e^-h_tI and gives strong convexity modulus µ=e^-h_t.This holds for every data law without assuming convexity of −log p_t, using the posterior covariance representation.
  • Solver behavior: Picard iteration is unit-step gradient descent, so oscillation and divergence reflect solver stability rather than multiplicity or model geometry.Damping η<2/Λ_t cures instability, with η⋆=2/(µ+Λ_t) optimal; an exact Bayes example oscillates despite a unique minimizer.
  • Model-error certificates: The unconditional ceiling σ_tλ_max(sym J)≤1 follows from posterior covariance positivity and is violated by trained DDPM scores in the tested image settings.At Bayes, equality occurs when the extremal posterior covariance direction is singular; trained violations certify over-stiffness.
  • Root uniqueness: The Bayes-limit solution is unique, while a second trained-model solution requires σ_t+1λ_max(sym J_ϕ)>1/(1−e^-h_t).The condition is necessary but not sufficient because it bounds only the symmetric Jacobian part.
  • Geometric convergence domain: The scale-free oscillation boundary is w_osc=1/2, while the divergence boundary is w_div=1/(1+ρ_g^⋆); finite-noise corrections depend quadratically on ||II||².The two-term curvature correction matches all 27 scanned settings, whereas the one-term form misses 8 of 27.

C DEFERRED METHOD MATERIAL

The deferred material calibrates score-based geometric readouts, identifies estimator ceilings and biases, and limits interpretation through reach, codimension, and probe-size constraints.

  • Bayes calibration: Analytically integrable manifolds provide machine-precision Bayes values for the paper’s readouts across noise levels and codimensions.The calibration uses a circle in R², a sphere in R³, and a circle in R³ with closed-form Gaussian convolutions.
  • Dimension readouts: The noiseless-point readout returns intrinsic dimension d, whereas the same functional at sampled noisy points returns ambient dimension D.The distinction is formalized by Proposition 7 and Proposition 6.
  • Readout relationships: The Fokker–Planck and DSM readouts differ by an exact optimality-gap term, so they cannot corroborate one another when that gap is nonzero.The gap is the orthogonality defect of the trained denoiser against the Bayes denoiser.
  • Estimator ceilings: Local PCA and score-rank estimators are capped by the number of score or projected samples, making them nonbinding at the prescribed K=4D.Centring reduces the score-evaluation span bound by one degree of freedom.
  • Scope boundaries: Reach and probe-subspace conditions constrain geometric interpretation, including the requirement that a random probe dimension exceed D−nd.At a free boundary, an outward tangential direction can be truncated and read as normal.

D DEFERRED DERIVATIONS

The deferred derivations establish exact probe identities, validity windows, product factorization, and rank-readout behavior, including protocol-dependent failures on trained scores.

  • For a linear manifold, the exact DSM readout satisfies L_t/ᾱ_t = d at the scaled clean point.
  • The Bayes residual contains a curvature term, requiring r ≪ σ_t^2∥H∥/2 for curvature readouts.At CIFAR noise σ_t = 0.0264, this requires r < 3.5 × 10^-4, or relative positional accuracy 6 × 10^-6.
  • For product manifolds, reach(M) = min_p reach(M_p), while additive readout bias is independent of the number of factors.
  • The Fermi and support conditions jointly limit observable depth to w ≤ 0.14, 3.6× below the oscillation shell and 5.6× below the divergence shell.Both conditions hold in 0 of 25 settings; σ_t+kκ_max spans 0.059 to 3.712 against the 0.10 Fermi window.
  • At the prescribed rank-estimator budget, exact scores recover codimension on four of five classes by the ratio rule and five of five by the difference rule, while trained settings fail.The ratio rule abstains on trained settings, whereas the unthresholded difference rule emits a confident ambient-dimension estimate.

E FROM THE SMALE SURROGATE TO THE KANTOROVICH TRIPLE

The paper replaces an uncomputable Smale surrogate with a measurable Kantorovich triple and verifies the resulting certificates, while showing that undamped Picard boundaries remain exact.

  • The Smale surrogate is invalid because its second-order bound fails at the Bayes limit and its higher-cumulant constant lacks a closed form.For piecewise-linear blocks, the surrogate constant is infinite; the unconditional Hessian inverse bound is instead ∥(∇²Ψ_t)^−1∥ ≤ e^h_t.
  • Kantorovich’s theorem uses measurable first derivatives and a local Lipschitz constant, with Bayes references for all three quantities.
  • 20 of 20 point-cloud settings and 4 of 4 CIFAR-10 settings satisfy the Kantorovich certificate.Exact-score point-cloud values range from 4.64 × 10^-3 to 8.54 × 10^-2; trained values range from 8.89 × 10^-4 to 2.83 × 10^-3.
  • Tangent projection is necessary for point-cloud Lipschitz estimates, where random directions understate L by factors of 5 to 21.On CIFAR-10, the rank-1 projection is effectively inert, with L_rand/L = 0.9995–1.0001.
  • The exact Picard boundary is w_P = 1/(1 + ρ⋆_g), verified to 1.3 × 10^-4, whereas the earlier α-theory constant is withdrawn.

F PRIOR WORK, CLAIM BY CLAIM

The prior-work review distinguishes inherited identities and bounds from the paper’s contributions: explicit potential and constants, estimator-validity readings, and checkable certificates.

  • The paper claims no novelty for several antecedents, including the Tweedie–Miyasawa identity and prior model-geometry interpretations.
  • Its contribution is a post-hoc calibration framework for arbitrary pretrained scores, including the convergence-domain law and the covariance-derived ceiling.
  • The dimension functional is evaluated exactly for every model and data law, explaining its behavior rather than merely bounding it.
  • The explicit potential Ψ_t, modulus e^-h_t, and factorized contraction constant ρ_g = ρ⋆_g S_t separate schedule effects from model effects.
  • The topological obstruction to metric repair applies to closed positive-dimensional manifolds, but not necessarily contractible patches or the swiss roll.
  • The radial-index certificate establishes existence without contraction, reach, manifold, or unimodality assumptions.It fires in 24 of 24 CIFAR and 27 of 27 CelebA-HQ settings.

G DEFERRED EXPERIMENTAL DETAIL

Deferred experiments show that exact scores reproduce calibrated exponents and convergence shells, whereas trained scores fail dimension and shell tests for structurally interpretable reasons.

  • Exact scores recover codimension on closed classes, but trained models read approximately the ambient dimension and fail to separate d = 1 from d = 2.The epoch-300 model reads 382–383 on five classes, while the exact rank statistic recovers the target classes.
  • For the scale-free dimension test, trained ratios are near 1.0003, versus 0.496–0.536 for exact scores across 12 guarded pairwise settings.The exact score places the true one 16 null standard deviations below the others; the trained model places it 0.1 above.
  • The descent exponent is a landscape-quadraticity detector rather than a reliable estimator: it returns 0.5000 under a divergent step and 0.077 on a quartic energy with true exponent 3/4.The field estimator returns the true exponent on all four analytic energies.
  • At the CIFAR probe, trained Picard rates are 19%–27% above the Bayes contraction reference, consistent with σ_tλ_1(J) > 1.At stride 20, empirical Picard divergence occurs in three images while the Bayes map contracts at 0.7134.
  • Exact-score oscillation shells occur near w = 1/2, with the torus’s finite-noise correction matching prediction to 0.8%.Variable-curvature classes fall outside the single-κ_max scope rather than contradicting the law.
  • No trained shell is observed because the probe’s Fermi and support windows are structurally incompatible by factors of 3.6–5.6.The conditions are jointly satisfied in 0 of 25 convergence settings.

H FALSIFIABLE PREDICTIONS AND FAILURE ATTRIBUTION

The run separates estimator, model, budget, window, and untested failures by comparing trained readings with exact or oracle controls under matched settings. Several predictions are confirmed, while others are narrowed or remain unresolved.

  • Attribution: The exact-score control shows flattened readouts cannot separate d=1 from d=2, whereas the trained score never lowers the same ratio below 0.996.The exact score reaches 0.505 under the identical budget, establishing a structural trained-score failure rather than a readout failure.
  • H1: 51 of 55 settings close the trained-to-exact score gap across training, but monotonic closure occurs in only 3 of 55 settings.The four exceptions occur at t=64, where the model already overshoots the exact score at epoch 5.
  • Attribution: Table 5 labels each failure as estimator, model, budget, window, or untested, with exact controls using the identical harness and image entries treated as certified lower bounds.Hypothesis-free rows are not discounted as out of regime because their assumptions require no manifold, reach, or unimodality.
  • H1: 2.15/2.29/2.16/2.21/2.09× inward movement of the half-stiffness noise front occurs across five classes, but it is non-increasing at every step for only three.The supported form of H1 is therefore a front rather than per-setting monotonicity.
  • H4: 124 of 192 patch directions satisfy ν > 1, yielding d̂ = 68 or 35.4% of patch dimension; resolved-only and midpoint rules give different conclusions.The 36% dead band prevents one patch at one noise pair from separating the two readings under all rules.

WHAT THE RUN DID NOT SETTLE

The run leaves observability of the convergence-domain law, the multiplicity certificate on trained scores, and image-failure attribution unresolved. These limits reflect narrow measurements, absent trained-score shells, and a confound inside the training noise shell.

  • Convergence-domain law: Whether the convergence-domain law is observable on trained scores remains open, while the measured Hessian–Lipschitz constant is only 2–12% of the curvature it reads.This estimate is based on a two-point σ-fit on one network without an error bar.
  • Multiplicity certificate: The multiplicity certificate has a theorem-level threshold verified in an analytic model but zero observations on trained scores in this study.The supplied conclusion distinguishes the certificate’s theoretical status from its empirical observability.
  • Image failures: Image failures remain unattributed because posterior covariance along the score ray never exceeds 1.12σ_t^2, while the point forced by Prop. 6 lies D standard deviations inside the training noise shell.The live confound is the base point’s location rather than the unknown reach.
  • Experimental scope: The experiment uses two snapshots from one run and two networks from one recipe, so class-to-class comparisons have no population error bar.The networks differ in S_t by a median 3.0% and maximum 13.8%, reported as network spread rather than population uncertainty.
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