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

Preprint: Diffusion Inverse Step as a Strongly Convex Program

O NE I NVERSE S TEP IS A C ONVEX P ROGRAM : BAYES -L IMIT C ALIBRATION OF D IFFUSION I NVERSION Gordei Verbii Independent Researcher scigverbii@gmail.com

arXiv:2608.23094v1 [stat.ML] 24 Aug 2026

A BSTRACT 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) = ∇Ψt (x), strongly convex at the Bayes limit with modulus exactly e−ht for the step’s log-SNR gap ht — 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−ht ), a hypothesis-free certificate of model error; the same bound makes contraction a schedule constant, ρ⋆g = 1 − e−ht < 0.326 throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unitstep gradient descent on Ψt , unstable wherever λmax (∇2 Ψt ) > 2, so oscillation certifies nothing; damping below 2/λmax cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth w = rκmax the oscillation shell sits at w = 21 , schedule-free, and the divergence shell at w = 1/(1+ρ⋆g ), with a measured finite-noise correction in ∥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×, and the trained Hessian-Lipschitz constant is 2–12% of the curvature the law reads, 0 on a ReLU net. Finally the unconditional ceiling σt λmax (sym J) ≤ 1, from Cov(x0 | xt ) ⪰ 0 alone, holds for the exact score to 3 × 10−7 but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by 1.26–4.66×.

1

I NTRODUCTION

The manifold hypothesis holds that natural data x0 ∈ RD concentrate near a submanifold M ⊂ RD of local dimension d ≪ D (Pope et al., 2021). Diffusion models estimate the score of noise-convolved data (Ho et al., 2020; Song et al., 2021b) and so encode the local normal bundle, and a productive line of work reads local intrinsic dimension (LID) off pretrained models (Stanczuk et al., 2024; Kamkari et al., 2024; Horvat & Pfister, 2024; Yeats et al., 2025). The deterministic DDIM map (Song et al., 2021a) makes the cheapest such experiment available: solve one implicit fixed point, differentiate the network once, read the geometry off the Jacobian. This paper asks what that probe is a measurement of, and observes that the question has a closed-form answer. Were the network the Bayes-optimal denoiser of a d-manifold, every quantity the probe reports would take a value computable in advance from (d, D, σt ) and the mean curvature. That converts the probe from an estimator into an instrument with a calibration curve, and supplies its validity conditions for free: each readout is meaningful only inside a window of noise scales, bounded above by the reach and below by the model’s own calibration. The act of solving for it has a closed-form answer too, and that is what organises this paper. One implicit inversion step is the stationarity condition of an explicit potential, strongly convex at the Bayes limit with a modulus fixed by the schedule alone, which separates three objects the literature runs together. The solution set is a single point, unconditionally, so a second solution certifies model error with no geometric hypothesis. The solver’s orbit can nonetheless be pathological at the Bayes limit — Picard’s step size is pinned at 1 by the algebra of the map, not chosen — so oscillation and non-convergence certify nothing. And the geometry the probe was built to find is in neither: it is in 1

Preprint: Diffusion Inverse Step as a Strongly Convex Program

the size of the convergence domain, whose boundary sits at a fixed fraction of the manifold’s focal radius, set by the schedule alone. Contributions. 1. One inverse step is a convex program (Thm. 1): F = x − G(x) = ∇Ψt with 2 ∇2 Ψt = e−ht I + ρ⋆g Ct+1 /σt+1 ⪰ e−ht I, so the solution is unique for every data law and every schedule, Picard is unit-step gradient descent, and the condition number is eht — independent of the data, of d, of D and of curvature. 2. A contraction theorem, with its constant (Thm. 4): ρg ≤ ρ⋆g = |Bt |/σt+1 = 1 − e−ht < 1 for every schedule with ᾱT > 0 under Assumption 2, the constant itself being unconditional and below 0.326 on the standard DDPM schedule, and ρg = ρ⋆g St+1 separating schedule from model. 3. The convergence domain is a curvature (Thm. 5): both Picard stability boundaries are fixed fractions of the manifold’s focal radius, wosc = 1/2 schedule-free and wdiv = 1/(1 + ρ⋆g ), 2 with the measured finite-noise correction 12 σ̃t+1 ∥II∥2 — free of the data law, the density, the dimension and the codimension. Exact scores put both boundaries where the theorem does; on trained scores they are structurally unmeasurable by this probe, and we give the arithmetic. 4. A Bayes-limit calibration table (§5.1) giving the exact value of every readout of a one-step probe, verified on three integrable manifolds to machine precision in codimension 1 and 2 (Table 1), with two corrections√ to values assumed in this literature: the Bayes residual at the noiseless point is (σt /2)∥H∥/ ᾱt under a uniform measure, not 0 — also obtained, with a proof, by Chen et al. (2026, Thm. 3.4) — and the tangential scaling exponent is −1, not 0. 5. Three hypothesis-free falsifications, with an oracle control (§5.2) that separates a broken estimator from a broken model: σt λmax (sym J) ≤ 1, skew Q = 0 and σt tr J ≤ D follow from Ct ⪰ 0 alone; the exact score attains the first to 3 × 10−7 and the second to 2 × 10−6 , and the trained scores cross the first in 10 of 110 exactly-computed cloud settings and 45 of 45 image settings. 6. A negative result about a positive result (§5.2): the descent-based estimator of the Łojasiewicz exponent returns 12 with R2 = 1 whenever one scale governs the fit window, in particular under the divergent step the published protocol implies. We give the field-based estimator that does carry information, calibrate it, and measure it on CelebA-HQ-256.

2

R ELATED W ORK

Reading geometry off a score. Stanczuk et al. (2024) prove that at low noise the score points orthogonally toward M and that the score matrix has rank D − d; Kamkari et al. (2024) derive the Fokker–Planck estimator, Horvat & Pfister (2024) analyse conservativity, Yeats et al. (2025) connect the DSM loss to LID, image estimates are due to Pope et al. (2021); Tempczyk et al. (2022), and Ventura et al. (2025); Lu et al. (2024) describe the spectral phases and the blow-up rate on M. The differential geometry is classical — reach and the eiconal property (Federer, 1959), the tube volume density (Gray, 2004), the ray structure of ∇2 dist2 (Ambrosio & Mantegazza, 1998) — as is the Łojasiewicz reading we correct in Rem. 10 (Witten, 1982; Łojasiewicz, 1963). All of it scores the probe as an estimator against a ground-truth dimension; we read it as an instrument and ask what it returns on an input whose answer is known, which is what makes the negatives below attributable. Inversion as a fixed point, and as an optimisation problem. DDIM (Song et al., 2021a) admits an implicit fixed-point inversion whose solvers fall into two families the literature does not separate. Picard (Garibi et al., 2024), Anderson (Anderson, 1965; Pan et al., 2023) and Newton–Krylov iteration solve one residual with one root — over 70 settings they agree to 1.7 × 10−4 of the local noise scale and every readout is solver-independent to 1.6 × 10−6 relative — whereas the regularised and averaged variants (Samuel et al., 2025) solve a different residual, whose root is displaced by up to 24.5 noise scales. Convergence of the Picard branch is studied by Hong et al. (2024); Hang et al. (2024) and the deep-equilibrium reading of sampling by Bai et al. (2021); Pokle et al. (2022); none writes a potential, and the contraction constant is left free. Reading the step as gradient descent on a strongly convex potential puts it in classical territory: existence and a certified radius follow 2

Preprint: Diffusion Inverse Step as a Strongly Convex Program

from Kantorovich’s theorem (Kantorovich & Akilov, 1982) with no contraction assumption, and the spectral-radius escape of Ostrowski (1973) is closed here, where J ⋆ is a posterior covariance and ρ(Bt J ⋆ ) equals its norm. Smale’s γ-theory (Smale, 1986; Wang & Han, 1989; Blum et al., 1998) needs a constant with no closed form even at Bayes, so App. E uses the Kantorovich triple instead. What is ours, and what is not. Four antecedents bound our claims and we claim novelty for none: Farghly et al. (2025, §4), whose normal block gives the 1/σt constant of (6) and who argue before us that the expressed geometry is a property of the model; the Tweedie–Miyasawa identity (7) (Miyasawa, 1961; Kadkhodaie et al., 2024); Wenliang & Moran’s σt2 -normalised reading of a trained spectrum; and the codimension-scaled window of Niyogi et al. (2011, Thm. 1). What is new is the potential and its modulus, the schedule constant ρ⋆g , the convergence-domain law with its finite-noise correction, the ceiling assembled from Ct ⪰ 0, the sampled-point value D − δt /2, the δt identity with its exact remainder, the Hamilton–Jacobi constant, and Assumption 2 made checkable. App. F sets this out claim by claim, and places Diepeveen et al. (2025) — whose Riemannian structure is complementary to ours rather than rival, the Fisher metric being affine in J (Raskutti & Mukherjee, 2015) — and Azeglio & Di Bernardo (2025).

3

P RELIMINARIES AND P ROBLEM S ETUP

Data x0 ∈ RD lie near a manifold M ⊂ RD of local dimension d and reach reach(M), and the DDPM forward process (Sohl-Dickstein et al., 2015; Ho et al., 2020) uses a decreasing schedule ᾱt ∈ (0, 1): √ √ σt := 1 − ᾱt , ε ∼ N (0, ID ). (1) xt = ᾱt x0 + σt ε, pt is the law of xt and ∇ log pt its score. The trained noise predictor is εϕ (x, t), its Jacobian J(x, t) := ∂εϕ /∂x, and ε⋆ (x) := E[ε | xt = x] the Bayes-optimal denoiser with Jacobian J ⋆ ; a star denotes Bayes optimality and nothing else. Two√evaluation points must be distinguished: the sampled point xt of (1), and the noiseless point x◦ := ᾱt x0 . We write s1 (A) ≥ s2 (A) ≥ · · · for singular values, H for the mean-curvature vector, κ for a principal curvature, K for the sample budget of a rank estimator, N for the number of probed inputs, n for the number of points in a point cloud, and Lt := E∥εϕ − ε∥2 ,

Πt := E∥εϕ ∥2 ,

δt := D − Πt − Lt .

(2)

θ is the Łojasiewicz exponent of the score energy Et := 12 ∥εϕ ∥2 and νj a scaling exponent. Differentiating pt under the integral gives the Tweedie–Miyasawa identities (Robbins, 1956; Miyasawa, 1961; Efron, 2011) √ ε⋆ (x) = −σt ∇ log pt (x), E[ ᾱt x0 | xt ] = xt − σt ε⋆ (xt ). (3) √ The schedule, and the inversion fixed point. Write λt := log( ᾱt /σt ) for the log-SNR, ht := p √ λt − λt+1 > 0 for the gap of the step t+1 → t, and σ̃t := σt / ᾱt . With At := ᾱt+1 /ᾱt and Bt := σt+1 − At σt , the identity At σt = e−ht σt+1 gives, with no expansion in the step size, ρ⋆g :=

|Bt | = 1 − e−ht , σt+1

c := eht − 1.

(4)

Every schedule constant below depends on ht alone, so a translation of the log-SNR axis — VP against VE, SD3/Flux against DDPM — changes none of them. Inverting the generative step t+1 → t from a target y = xt exactly means solving x = G(x) := At y + Bt εϕ (x, t+1)

(5)

for x = xt+1 , by Picard (Garibi et al., 2024), Anderson (Anderson, 1965; Pan et al., 2023) or Newton-type (Samuel et al., 2025) iteration. The network is evaluated at the unknown, noisier latent, at index t+1: that index makes the residual a gradient field (§4.2), and every constant we quote uses it. Substituting εϕ (x, t) gives a map whose fixed point is not the DDIM preimage and whose Lipschitz factor |Bt |/σt = At c is unbounded in the stride. Assumption 1. M is a closed C 2 submanifold, p0 has a C 1 density ρ0 on M bounded away from 0, and σt < reach(M), so the normal tube is embedded and Fermi coordinates exist (Federer, 1959; Gray, 2004). 3

Preprint: Diffusion Inverse Step as a Strongly Convex Program

4

P ROPOSED M ETHOD

Throughout this section εϕ = ε⋆ and Assumption 1 holds, except where a statement says otherwise: Thm. 1 and Prop. 3 need no geometric hypothesis at all, which on natural images makes them the only readings we can trust. §5 verifies each identity numerically and measures the deviation on trained models. 4.1

T HE NORMAL - BUNDLE LAW AND A STIFFNESS CERTIFICATE

√ Write r = dist(x, ᾱt M), let ν be the outward unit normal and P the tangential projector. In Fermi coordinates the tube volume density is det(I − rAν ) with Aν the shape operator (Gray, 2004), so a Laplace expansion of pt gives ε⋆ (x) = (r/σt )ν − σ2t H + O(σt r) and J ⋆ (x◦ , t) =

I −P σ | {zt }

normal block: the 1/σt stiffness of the noise tube

+

O(σt ) | {z }

.

(6)

tangential block: curvature and density

Differentiating (3) once more gives the second-order Tweedie–Miyasawa identity: exactly, and for every p0 , 1 Ct (x)  J ⋆ (x, t) = , I− σt σt2 (7)
√ Ct (x) := Cov ᾱt x0 xt =x ⪰ 0, the form stated for the variance-exploding parametrisation by Kadkhodaie et al. (2024, Eq. (9)); we claim no novelty for the identity, only for the singular-value reading. Since s1 (J ⋆ ) = σt−1 maxi |1 − ci /σt2 | for ci the eigenvalues of Ct , s1 (J ⋆ ) = 1/σt exactly whenever the extremal direction carries ci = 0 — true of the D − d normal directions of an affine M (Farghly et al., 2025, Eq. (10)). On a curved M that variance is positive and s1 (J ⋆ ) < 1/σt strictly; conversely s1 (J ⋆ ) > 1/σt as soon as some ci > 2σt2 , which Assumption 1 does not exclude: on the unit circle with ρ0 ∝ e23 cos θ and σt = 0.2 < reach(M) = 1 the Bayes value at the antipode is s1 (J ⋆ ) = 48.6 against 1/σt = 5. We therefore add Assumption 2 (Posterior unimodality at the probe). λmax (Ct (ξ)) ≤ 2σt2 for every ξ on the Picard segment. This is a checkable curvature condition, not an opaque constant: inside the tube of a uniform M the exact Fermi value is λmax (Ct ) = σt2 /(1 − rκmax ), so it holds iff r ≤ 1/(2κmax ), half the focal radius, and with margin 2 for every log-concave law (Brascamp & Lieb, 1976). It excludes loss of log-concavity, not proximity to the medial axis — the counterexample above sits on M, at r = 0 — and it is not a proof convenience but exactly the no-oscillation condition of the solver (Thm. 5), of which it is the coarse-schedule limit, conservative by 2/(1 + ρ⋆g ) = 1.51 at the first step of the standard DDPM schedule. This motivates the dimensionless
St := σt s1 J(x◦ , t) , (8) σ2 S⋆t = 1 − t min ∥Aν ∥2F + O(σt4 ), 2ᾱt ∥ν∥=1 computable from one power iteration, with Aν the shape operator in the normal direction ν. Its Bayes value is not 1 at finite noise: the deficit is a curvature term and also the expansion parameter, so with gt := 1 − S⋆t the residual is O(gt2 ) and (8) is exact on S 2 . It is a Bayes reference only inside a two-sided window — gt ≤ 0.1 and σ̃t ≤ 12 reach(M), an edge gt cannot see since gt ≡ 0 in codimension 2 — which voids 34 of our 55 exact-score settings, where we abstain rather than quote a reference out of regime; and for a flattened product the reference takes the minimum of ∥Aν ∥2F over the probe points, not its mean (App. H). It vanishes whenever some normal direction has vanishing shape operator, automatic in high codimension — at D=3072 for every d ≤ 76 — so the deficit is measurable only in low codimension (App. C). We report St against S⋆t , never against 1. 4

Preprint: Diffusion Inverse Step as a Strongly Convex Program

4.2

O NE INVERSE STEP IS A CONVEX PROGRAM

Theorem 1 (The inversion potential and its modulus). Let εϕ = ε⋆ . The residual of (5) is a gradient field, F (x) := x − G(x) = ∇Ψt (x), Ψt = 12 ∥x − At y∥2 + Bt σt+1 log pt+1 (x), and for every data law, every schedule and every x, ∇2 Ψt = I − Bt J ⋆ (x, t+1) Ct+1 (x) ⪰ e−ht I. = e−ht I + ρ⋆g 2 σt+1 So Ψt is strongly convex with modulus exactly µ = e−ht = 1 − ρ⋆g , and Picard iteration is gradient descent on Ψt with step size exactly 1. Inside the tube of a uniform M the spectrum of ∇2 Ψt is {1 (×d), e−ht (×D−d)}: the condition number is eht , and smin (I − Bt J ⋆ ) = e−ht exactly. (Proof: App. B.) Three readings follow at no cost. Picard’s step size is pinned at 1 by the algebra of G, so its failures are step-size failures, cured by damping below 2/Λt and optimally at η ⋆ = 2/(µ + Λt ), where Λt := λmax (∇2 Ψt ) throughout (not the DSM loss Lt ). The conditioning of one inverse step is eht , set by the gap between the indices and not by the data, d, D or curvature. And the distance to illposedness is a schedule constant, so the Kantorovich constant a = ∥(∇2 Ψt )−1 ∥ ≤ eht (Kantorovich & Akilov, 1982) is an equality in the normal block — measured a/eht = 0.9947 on ten exact-score settings — where the Neumann bound (1 − ρg )−1 is loose and needs Assumption 2 besides. Theorem 2 (Uniqueness at Bayes, and the multiplicity certificate). For every data law and every schedule with ᾱT > 0 the fixed point (5) is unique at the Bayes limit, and a trained score admits a second solution only if
1 1 σt+1 λmax sym J(ξ, t+1) ≥ ⋆ = ρg 1 − e−ht for some ξ on the segment. A fold is then the only available bifurcation, and the roots born at it carry multiplier ρ⋆g σt+1 λmax > 1, so they are repelling whenever they exist: multiplicity is a Newton observable and is invisible to Picard. (Proof: App. B.) On a two-atom Bayes law with an amplitude-scaled score εϕ = γε⋆ the root count moves from 1 to 3 exactly where predicted, bracketed at ρ⋆g γ ∈ (0.996, 1.050] (App. B). That is a threshold verified in an analytic model; no trained score in this study was tested against it. Proposition 3 (The
unconditional ceiling). For every data law, every t and every x, σt λmax sym J ⋆ (x, t) ≤ 1, with equality iff Ct (x) is singular in the extremal direction. No manifold, reach, unimodality or tube hypothesis is used. Proof. J ⋆ = σt−1 (I − Ct /σt2 ) by (7) is symmetric, and Ct ⪰ 0. This is the only reading of a score Jacobian here that carries
no hypothesis at all, and it is why we report the signed pair σt λmax (sym J), −σt λmin (sym J) beside St : s1 fuses the ceiling branch with a medial-axis branch that has none, and on the von Mises circle above the Bayes value is St = 9.7 while the ceiling still holds. So St > 1 does not falsify Bayes optimality, while σt λmax (sym J) > 1 does. 4.3

O NE INVERSE STEP : A SCHEDULE - DETERMINED CONTRACTION

The mean-value inequality applied to (5) gives the Lipschitz constant |Bt | supξ s1 J(ξ, t+1) over the Picard segment; we estimate it at the probe point and call it ρg , and write ρemp for the measured per-step rate, so ρemp ≤ ρg . Power-iterating an implicit layer’s input Jacobian as a contraction certificate is the standard deep-equilibrium instrument (Bai et al., 2021; Pokle et al., 2022); what is new is not the instrument but its reference value, which (7) supplies in place of the Banach threshold 1. 5

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Theorem 4 (Bayes contraction, with a schedule-only constant). For the inversion map (5), under Assumptions 1 and 2 with d < D, ρg ≤ ρ⋆g =

|Bt | = 1 − e−ht < 1 σt+1

for every schedule with ᾱT > 0, every stride and every parametrisation, the bound being approached only as ht → ∞. Hence ρg = ρ⋆g St+1 identically, so the bound is attained exactly when St+1 = 1, i.e. when M is flat in some normal direction. A measured ρg ≥ 1 is an over-stiffness of the trained score rather than a property of one-step inversion, and it is a property of Picard alone: damping with η < 2/Λt converges at every ρg , at rate tanh(ht /2) when optimal inside the tube, and Newton and Anderson are insensitive to Λt . (Proof: App. B.) Evaluating ρ⋆g = 1 − e−ht at every step gives maxt ρ⋆g = 0.326, attained at the first step, on the standard DDPM schedule (T =1000, β linear on [10−4 , 0.02]): one-step inversion is contractive at every step in the Bayes limit, and unconditionally. The coarse T =100, βend =0.05 schedule reads 0.623, not the 1.65 of |Bt |/σt = At c, which belongs to the explicit map and has no fixed point to contract to. The bound is attained only as ht diverges: a zero-terminal-SNR top step reads exactly 1, and a cosine schedule binds at its last step at a value set by the library’s ᾱ floor (0.968 or 0.936) rather than by the schedule. The other five families satisfy the identity to 9.4 × 10−16 and stay strictly below 1 at strides 1 to 100, with ρ⋆g spanning 0.024 to 0.971: the headroom is a design choice, not a constant of the method (App. A). 4.4

T HE CONVERGENCE DOMAIN IS A CURVATURE

Theorem 5 (Focal-radius law, at the Bayes limit). Let Assumption 1 hold, let εϕ = ε⋆ , and displace inward from x◦t+1 along a principal normal by r, writing wi := rκi and w := rκmax for the scalefree depth (1/κmax being the focal radius). The Picard multipliers are then exactly +ρ⋆g (× D−d) and −ρ⋆g wi /(1 − wi ) (× d), for every σt and every data law. Hence (i) the oscillation boundary, where the dominant multiplier changes sign and equivalently where Assumption 2 fails, is wosc = 12 , independent of the schedule; (ii) the divergence boundary λmax (∇2 Ψt ) = 2 is wdiv = 1/(1 + ρ⋆g ); and (iii) at finite noise both carry a closed-form correction set by the full second fundamental form, 1 2 wdiv = + 21 σ̃t+1 ∥II∥2 + O(σ̃ 4 ), 1 + ρ⋆g (9) wosc = 21 − 41 (σ̃κ)4 + O(σ̃ 6 ). (Proof: App. B.) Both boundaries are fractions of the focal radius fixed by the schedule alone: on DDPM-linear at strides 1/10/20/50/100, wdiv = 0.754/0.560/0.534/0.515/0.507 while wosc = 12 at every one. The same expansion gives ∇2 Ψt a ray-Lipschitz constant LH (w) = ρ⋆g κmax /(1 − w)2 , so with a ≤ eht the Kantorovich product is c κmax /(1 − w)2 and no certified ball reaches the focal point. The multipliers are exact at every σt and every data law, so both shells are recoverable from a matrix-pencil fit to the Picard error sequence with forward passes only; only (9) is an expansion. Two scope clauses belong with it. The scope is the Bayes limit and the oracle: §5.2 finds both shells on exact scores and none on any trained score, for a reason measured there. And the displacement must be inward: an outward ray reads λmax (Ct )/σt2 = 1/(1 + w), crosses no threshold and returns a confident null — on the exact score it resolves both branches at all 18 depths and reports no shell, while its inward twin reads a multiplier 8.66× larger. 4.5

F OUR FURTHER READOUT IDENTITIES

The remaining readouts have exact Bayes values too, each stated with its proof and attribution in App. C: the transverse Hamilton–Jacobi law (13), fixing the Łojasiewicz exponent of the score energy at 12 on the whole critical manifold and supplying its constant, with Rem. 11 on why a descent estimator returns it tautologically; the scaling exponents (14), split ±1 and not {+1, 0}; the identity δt = D − Πt − Lt (Prop. 8); and the rank ceiling (Prop. 9). One of them decides where a probe may be evaluated at all. 6

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Table 1: Bayes-limit calibration; no trained model enters this table. Above the rule every entry is a dimensionless ratio with Bayes value 1 (or ±1 for ν). Below it are the identities that make the table a calibration rather than a list, and the sweep that shows it resolves. M

σt

St /S⋆t

√ 2 ᾱt ∥ε⋆ (x◦ )∥ σt ∥H∥

ˆ ◦) d(x ˆ d⋆ (x◦ )

ν⊥ /ν∥

S 1 ⊂ R2 , d=1

0.10 0.02 0.10 0.02 0.10 0.02

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.003 1.000 1.000 1.000 1.003 1.000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

+1.02/ − 1.06 +1.00/ − 1.00 +1.05/ − 1.04 +1.00/ − 1.00 +1.01/ − 1.06 +1.00/ − 1.00

S 2 ⊂ R3 , d=2 S 1 ⊂ R3 , d=1

µ(∇2 Ψt )eht , ρ⋆g /(1−e−ht ) 2wosc , wdiv (1+ρ⋆g ) order in σt κ: residual, dˆ

1 ± 2·10−16 , 1 ± 9.4·10−16 1 − 3.1·10−10 , 1 + 1.7·10−5 2.004–2.013, 4.006–4.017

Proposition 6 (Value at the sampled point). Let p0 have compact support or sub-Gaussian tails and let εϕ be weakly differentiable with pt εϕ → 0 at infinity. Then the Fokker–Planck dimension functional dˆ = D − σt tr J + ∥εϕ ∥2 of Kamkari et al. (2024) satisfies, for every such p0 and every ˆ t )] = D − 1 δt , with δt the optimality gap of (2); at Bayes, exactly D. model εϕ , Ext ∼pt [d(x 2 Averaged over sampled points the readout is an affine function of the model’s orthogonality defect and of nothing else, so it reports the optimality gap and nothing about the manifold: the noiseless point is forced rather than chosen, and the bound Ex̃ [FLIPD] ≥ d of Yeats et al. (2025, Eq. (7)) is not merely loose — its exact value is D. At x◦ , where Leung et al. (2025, Thm. 1) prove the limit is d − D, √ it does recover the manifold, with a signed curvature correction and a Bayes residual (σt /2)∥H∥/ ᾱt rather than 0 (Prop. 7).

5

E XPERIMENTS

5.1

C ALIBRATION ON I NTEGRABLE M ANIFOLDS

Each Bayes value above is a prediction checkable with no trained model. For the uniform measure on a circle in R2 , a sphere in R3 and a circle in R3 the Gaussian convolution is closed-form (Bessel I0 , sinh, I0 ), so ε⋆ , J ⋆ and ∇2 Et are analytic to machine precision (App. A). Table 1 runs every readout on them at five noise levels in codimension 1 and 2, with the identities the same suite must satisfy, and Fig. 2 plots the sweep. The readouts are not intrinsically broken: on an exact score they match their closed forms to machine precision in codimension 1 and 2 — (8) is exact on S 2 , not merely O(σt4 ) — so whatever fails on trained models is a property of those models or of the protocol. Because the table then reads 1.0000 throughout, the evidence that it resolves anything is its last row: over a factor 4 in κ and a decade in σt the two ratios leave 1 at the predicted orders 2 and 4. And θ = 21 with R2 = 1.000000 appears on all three manifolds regardless of d or topology, exactly as (13) requires, which already shows the exponent cannot discriminate geometry. 5.2

M EASUREMENTS ON T RAINED M ODELS

Protocol. We probe (i) pretrained DDPM CIFAR-10 (D=3072, N =4) and DDPM CelebA-HQ256 (D=196 608, N =3), both fp32; (ii) a conditional score model over five synthetic point clouds (n=128 points in R3 , D=384) at seven training snapshots; and (iii) the exact Bayes denoiser of each cloud, run through the identical harness, probe cloud, flattening and budget. That third arm separates a broken estimator from a broken model and is what makes every negative below attributable. Three disclosures: the point-cloud probes are conditional; both image models are probed at model samples rather than dataset images, so image St is an upper estimate; and the two point-cloud states are epoch-300 and epoch-50 snapshots of one run. With N ≤ 4 we report individual values, not intervals. 7

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Stiffness and contraction. The unconditional half of the signed pair settles first. Because Ct ⪰ 0, σt λmax (sym J) ≤ 1 for every data law at every noise level — no manifold, no reach, no unimodality (Prop. 3) — so a reading above 1 is a hypothesis-free certificate of model error, the one measurement here needing no validity window. The exact score attains that ceiling and never leaves it, the largest of 55 exactly computed cloud spectra being 1.0000003. Every trained score we probe crosses it: 10 of 110 cloud settings up to 1.1323, on four of five classes and in both training states; 36 of 36 CIFAR-10 settings by 1.2586 to 2.4119; 9 of 9 CelebA-HQ-256 settings by 1.6146 to 4.6637. Since a shifted power iteration returns ∥Av∥ on a positive-semidefinite operator, every image value is a certified lower bound on its own excess. Two readouts of the same kind agree (Table 5), and the skew index carries a consequence: exactly 0 at Bayes because J ⋆ is a posterior covariance, it puts up to 43% of s1 (J) into the antisymmetric part on CIFAR-10 against 2·10−6 on the exact score, so St is not readable as a stiffness on images at all. The signed pair is a necessity, not an improvement. Where St is readable it costs one power iteration and is, by Thm. 4, the entire explanation of the measured contraction rate. At the smallest guard-clean cloud setting (σt =0.0426) the exact score reads St /S⋆t within 8·10−8 of 1 on the two closed, uniformly sampled classes, while the epoch300 network reads 0.050–0.055 and the epoch-50 one 0.024–0.034: an 18 to 42× deficit against a reference computed in the same harness at the same setting. At the CIFAR probe ρ⋆g = 0.1180 and the measured Picard rate is 0.1409–0.1498 — one-sided high, exactly what ρg = ρ⋆g σt λ1 (J) with σt λ1 > 1 predicts — and at stride 20 Picard measurably diverges where the Bayes map contracts at 0.7134 (App. G). None of this is evidence for the Bayes law: on an affine M the normal block of the noised empirical log-density is data-independent (Farghly et al., 2025, Eq. (10)), so a perfectly memorising score also reads St = S⋆t , and the certificate is one-sided. What the flattened dimension readouts measure. A point-cloud sample is n points in R3 flattened to D=3n, and the estimators of §4.5 measure dim supp pt in the ambient space they are run in. Because εϕ acts per point, ∂(εϕ )p /∂xq = 0 for p ̸= q, so the probed law is a product whose support has dimension nd: the flattened target is nd ∈ {128, 256}, not d ∈ {1, 2}, and comparisons against d are mis-specified. At the prescribed K = 4D = 1536 no rank readout on a trained model survives: the largest consecutive gap ratio is 1.02–1.58 over all ten trained settings against a pure-noise null of 1.022 at the same (K, D), and every one abstains — including the sphere, whose codimension-128 gap ratio of 5.06 at K = 1.08D falls to 1.16 at index 373. That reading was a budget artefact: H5 is confirmed as stated — rank readouts do track their budget — and refuted as a hope, since raising K to protocol retired our own positive along with the negative it was meant to rescue (App. H). The oracle control. Replacing εϕ by the exact score under the identical harness, probe cloud, flatˆ ◦ ) = 255.99999, 255.99999, 256.99, 246.32, 128.16 on tening and budget at σt = 0.0426 gives d(x the sphere, two spheres, torus, swiss roll and trefoil against flattened targets 256, 256, 256, 256, 128; the rank statistic recovers the true codimension on every closed class and the exponent split of (14) on four of five. The epoch-300 model reads 382–383 on all five classes and the epoch-50 one 391–410, i.e. the ambient D: the failure is a property of the trained score, not of the readout, and both oracle misses are the swiss roll’s free boundary (App. G). A negative that no change of target repairs. The decisive test needs no ground truth: any hypothesis of the form “the readout tracks nd” predicts that the d=1 manifold reads exactly half of the d=2 manifolds. Over the 12 pairwise settings that satisfy every validity guard — the trefoil against each two-dimensional class separately, never against a mean — the epoch-300 network’s ratio lies in [0.996, 1.008] with mean 1.0003, against an exact score reading [0.496, 0.536] on the very same settings; letting each class in turn play the d=1 role, the exact score puts the true one 16 null standard deviations below the others while the trained model puts it 0.1 above. Being a ratio it is invariant to the ground truth and to any common multiplicative bias. The flattened amplitude readouts carry no class information about d in this setting (App. G). The convergence domain, measured. On the exact score both shells sit where Thm. 5 puts them (Fig. 1a). At σt+1 = 0.0585 the oscillation shell reads wosc = 0.499946, 0.499997 and 0.497288 on the sphere, the two spheres and the torus against the schedule-free 12 , and the divergence shell wdiv = 0.789124 against 1/(1 + ρ⋆g ) = 0.785615, the residual +0.003509 being the predicted 1 2 2 2 σ̃t+1 ∥II∥ = 0.003440 — a two-term correction the torus confirms to 0.8%. The three classes 8

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Figure 1: The convergence domain, and why the trained shell is unmeasurable rather than absent. (a) Dominant Picard multiplier along an inward normal ray (sphere, σt+1 =0.0585): on the exact score the negative branch (dashed) takes over at wosc = 0.4999 and reaches 1 at wdiv = 0.7891, against the closed forms 12 and 0.7856; the trained field is flat and crosses neither. (b) The Fermi window (vertical) caps the reachable depth at 1.4 σt+k κmax (diagonal, one point per convergence setting), 3.6× short of wosc : the hatched box where both hold is empty by arithmetic (§6). reading 12 are exactly the three whose largest principal curvature is constant over the cloud; the other two, whose curvature spreads threefold, return a divergence cliff at wdiv /wosc = 1.05 where a Bayes pair must be separated by 1.571, so a single κmax is not the supremum the law asks for and they fall outside its scope rather than contradict it (App. G, which also excludes the trefoil a priori). On every trained score the same sweep returns no shell, and that is a measurement rather than an absence. Of our 88 trained shell readouts, 21 sit at a setting where the exact score does resolve one; all 21 are null, and the other 67 carry no information because the exact score is null there too. The mechanism is measured three ways, on two independent draws of one recipe. On the convergence network the trained Jacobian is 11.4–13.7× too weak in amplitude at the smallest setting; on the solver network, independently trained, the multiplier field is flat, at most 0.7113 over all 1491 admitted probes against a divergence threshold of 1, and the enabling lemma is absent, reading 0.021–0.124 against the exact score’s 0.92–0.995 when normalised by ρ⋆g κmax , and vanishing identically on a ReLU backbone. A score whose second derivative is a tenth of the curvature has a multiplier field that does not bend along the ray, so there is no shell to find — a negative that is scale-free and has a closed-form Bayes value on both branches. Three qualifications are underpowered rather than negative and App. H states each: a suppressed flip, a divergence shell refuted where the oscillation shell is strictly unresolved, and σt -exponents that are two-point slopes on one network. That no shell is found here is derived, not observed: the Fermi window and the model’s own training support cap the reachable depth at w ≤ 0.14 against shells at w ≥ 12 , jointly satisfiable in 0 of our 25 settings (Fig. 1b; §6). What the descent-based estimator measures. The estimator used in practice initialises at N (0, ID ), runs fixed-step gradient descent on Et and regresses log ∥∇Et ∥ on log(Et − mink Et ). By Remark 11 it must return 12 wherever the transverse spectrum is degenerate, whatever the model, and Fig. 3 confirms this on analytic energies with no diffusion model at all: θ̂ = 0.5000 with R2 = 1.000000 on the Morse–Bott energy under the divergent step the published hyper-parameters imply, and 0.077 on a genuinely quartic energy whose true exponent is 34 . It is a landscape-quadraticity detector with a mis-calibrated scale, not an estimator of θ; the field estimator returns the true exponent on all four (App. G). The image runs supply the counterexample. On pretrained CIFAR-10 and CelebA-HQ-256 — converged by any external standard — the descent protocol returns values indistinguishable from its own undertrained pole, and they are not merely inaccurate but inadmissible: a non-constant C 1,1 function admits no Łojasiewicz inequality with θ < 21 (Rebjock & Boumal, 2025, Rem. 2.21). The mechanism is the min-subtraction, which reproduces them with no model at all, so we do not report θ̂ as a training certificate. What (13) supplies instead is the field-based readout calibrated in §5.1: sample Et and ∇Et along a normal ray at x◦ , taking the slope as a test of transverse quadratic structure and the intercept as σt2 λ̂ → 1. On exact scores it returns θ = 0.50000 and σt2 λ̂ = 0.9999; 9

Preprint: Diffusion Inverse Step as a Strongly Convex Program

on pretrained CelebA-HQ-256, θ = 0.468–0.523 with R2 ≥ 0.9998 on two of three images, against a descent estimate below 0.01 on the same model, and along a random ray it does not (0.69–0.95). The intercept fails everywhere, so with N = 3 we report the slope and not the amplitude, and at t = 4 the fit is near-perfect at an inadmissible exponent — a falsification with a clean fit rather than noise (App. G). Validity windows. Every readout of §4 has a two-sided window in σt whose upper edge depends on where it is evaluated: at x◦ it is σt ≪ √ reach(M), while a sampling estimator evaluates at xt , whose √ distance to ᾱt M concentrates at σt D − d, a condition stricter by a factor 11 at D − d = 128 — not new (Niyogi et al., 2011, Thm. 1), but read here as an estimator-validity criterion rather than a hypothesis of a recovery guarantee. Its effect is to remove an excuse: all five classes are inside the window at the smallest probed noise once the product structure is accounted for, so the trefoil’s failure to separate d=1 from d=2 leaves the protocol column and the oracle control above places it in the model column (App. G). On images the upper edge additionally requires reach(M), which is estimable (Aamari et al., 2019; Yaguchi et al., 2025, Thm. 3.7, Alg. 1).

6

D ISCUSSION AND C ONCLUSION

Falsifiable predictions. Four of our five are decided, and App. H gives each verdict with its measurement. H1 is confirmed end to end — the gap to the exact score closes in 51 of 55 settings — and refuted as the fixed-t monotonicity we stated it as, which holds in 5 of 55; the correct form is a front, moving inward by 2.09–2.29×. H2 is confirmed on every class satisfying Assumption 1 with a uniform base measure. H4 is underpowered, the patch-wise exponent reading 68 of 192 with 36% of directions unresolved. H5 is confirmed as stated and refuted as a hope: raising K to the prescribed 4D moved artefacts in both directions, retiring our own positive rank reading with the negative it was meant to rescue. H3 was not run. Limitations. Assumptions 1–2 are unverified on natural images, so our image statements are calibration measurements rather than dimension claims: exactly four of the readouts we report on images are hypothesis-free — the ceiling σt λmax (sym J) ≤ 1, the skew index, σt tr J ≤ D and the sampled-point identity of Prop. 6 — and those are precisely the four that fire. Every reading that needs a reach, a Fermi window or a tube edge is uncheckable there. St in particular is one-sided and, on images, unsound — it fuses a positive branch that has an unconditional ceiling with a medial-axis branch that has none — so we report the signed pair throughout and keep St only where the skew index licenses it. The trained convergence shell is structurally unmeasurable by this probe, and that is a derived result rather than a null one. Three requirements collide. The shells sit at wosc = 12 and wdiv = 1/(1 + ρ⋆g ) ≃ 0.79. Converting a shell radius into a curvature needs the Fermi window σt+k κmax ≤ 0.1. And the probe must stay inside the model’s training support, r ≲ 1.4 σt+k . The last two bound the observable depth by w = rκmax ≤ 1.4 σt+k κmax ≤ 0.14 — short of the oscillation shell by 3.6× and of the divergence shell by 5.6×, for any manifold, any schedule and any training budget, because neither bound refers to the model. Measured: 0 of 25 trained settings satisfies both conditions, σt+k κmax spans 0.059 to 3.712 against a 0.10 window with one setting inside it, 18 of 25 satisfy even the looser r ≤ 2σt+k criterion of Table 3 alone, and in that one Fermi-admissible setting the predicted shells sit at r/σ = 8.5 and 13.4, six and ten times beyond the support. This is why every trained shell is reported against the exact score at matched noise and never against the closed form, and it is a statement about what the instrument can resolve — not a finding that the trained score has no convergence-domain structure. Three things we predicted and could not measure, each stated in App. H in the form we are willing to defend. Whether the convergence-domain law is observable on any trained score is open, and our own measurement of why it is not observable here — a Hessian–Lipschitz constant at 2–12% of the curvature the law reads — is a two-point σ-fit on one network with no error bar. The multiplicity certificate has a threshold verified in an analytic model and zero observations on a trained score anywhere in this study. And the image failures stay unattributable for a measured reason: along 2 the score ray the posterior covariance √ never exceeds 1.12σt , so no shell’s σ-exponent can be taken, and the point Prop. 6 forces sits D standard deviations inside the training noise shell — the live 10

Preprint: Diffusion Inverse Step as a Strongly Convex Program

confound, in place of the unknown reach. Our two training states are snapshots of one run and our two networks two draws of one recipe, so no trained class-to-class comparison here carries a population error bar. And we claim the focal-radius law for the Bayes limit and the oracle only: its multiplier statement holds at every σt and every data law, but the finite-noise correction and any radius-to-curvature conversion hold only inside the Fermi window, and the law is observable only where κmax is roughly constant over the probed set. Two cheap measurements remain: the field-based θ on a held-out image set, and a reach estimator on natural images — the prerequisite for any image LID claim including ours. Treating a one-step geometry probe as an instrument with a calibration curve changes what it can be asked. Every readout has a closed-form Bayes value, and verifying all of them against integrable manifolds converts each measured deviation into a statement about the trained model. Four consequences stand out. One inverse step is a strongly convex program whose modulus is a schedule constant, so its solution is unique and the solver’s failures are step-size artefacts to be damped rather than evidence about the model. The contraction rate is that same schedule constant. The Fokker–Planck dimension functional, averaged over sampled points, measures a model’s optimality gap and never its data manifold, which is why the noiseless evaluation point is forced. And the geometry the probe was built to find lives in the size of its own convergence domain — where an exact score reproduces both closed-form boundaries, no trained score in this study produces either, and the reason it cannot is an inequality we compute rather than a limit of our budget.

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Koichi Miyasawa. An empirical Bayes estimator of the mean of a normal population. Bulletin of the International Statistical Institute, 38(4):181–188, 1961. Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In Proceedings of the 38th International Conference on Machine Learning, volume 139 of PMLR, pp. 8162–8171, 2021. Partha Niyogi, Stephen Smale, and Shmuel Weinberger. A topological view of unsupervised learning from noisy data. SIAM Journal on Computing, 40(3):646–663, 2011. doi: 10.1137/090762932. Alexander M. Ostrowski. Solution of Equations in Euclidean and Banach Spaces. Academic Press, New York, 3rd edition, 1973. Zhihong Pan, Riccardo Gherardi, Xiufeng Xie, and Stephen Huang. Effective real image editing with accelerated iterative diffusion inversion. arXiv preprint arXiv:2309.04907, 2023. URL https://arxiv.org/abs/2309.04907. Ashwini Pokle, Zhengyang Geng, and J. Zico Kolter. Deep equilibrium approaches to diffusion models. In Advances in Neural Information Processing Systems (NeurIPS), volume 35, 2022. URL https://arxiv.org/abs/2210.12867. Phillip Pope, Chen Zhu, Ahmed Abdelkader, Micah Goldblum, and Tom Goldstein. The intrinsic dimension of images and its impact on learning. In International Conference on Learning Representations (ICLR), 2021. URL https://arxiv.org/abs/2104.08894. Garvesh Raskutti and Sayan Mukherjee. The information geometry of mirror descent. IEEE Transactions on Information Theory, 61(3):1451–1457, 2015. doi: 10.1109/TIT.2015.2388583. Quentin Rebjock and Nicolas Boumal. Fast convergence to non-isolated minima: Four equivalent conditions for C2 functions. Mathematical Programming, 2025. doi: 10.1007/s10107-024-02136-6. URL https://arxiv.org/abs/2303.00096. Herbert Robbins. An empirical Bayes approach to statistics. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pp. 157–163. University of California Press, Berkeley, CA, 1956. URL https://projecteuclid.org/euclid. bsmsp/1200501653. Dvir Samuel, Barak Meiri, Haggai Maron, Yoad Tewel, Nir Darshan, Shai Avidan, Gal Chechik, and Rami Ben-Ari. Lightning-fast image inversion and editing for text-to-image diffusion models. In International Conference on Learning Representations (ICLR), 2025. URL https://arxiv. org/abs/2312.12540. Steve Smale. On the efficiency of algorithms of analysis. Bulletin of the American Mathematical Society (New Series), 13(2):87–121, 1985. doi: 10.1090/S0273-0979-1985-15391-1. Steve Smale. Newton’s method estimates from data at one point. In Richard E. Ewing, Kenneth I. Gross, and Clyde F. Martin (eds.), The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, pp. 185–196. Springer-Verlag, New York, 1986. doi: 10.1007/ 978-1-4612-4984-9 13. Jascha Sohl-Dickstein, Eric A. Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In Proceedings of the 32nd International Conference on Machine Learning (ICML), volume 37 of Proceedings of Machine Learning Research, pp. 2256–2265, 2015. URL https://proceedings.mlr.press/v37/ sohl-dickstein15.html. Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In International Conference on Learning Representations (ICLR), 2021a. URL https://arxiv.org/ abs/2010.02502. Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations (ICLR), 2021b. URL https://arxiv.org/abs/ 2011.13456. 14

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Jan Pawel Stanczuk, Georgios Batzolis, Teo Deveney, and Carola-Bibiane Schönlieb. Diffusion models encode the intrinsic dimension of data manifolds. In Proceedings of the 41st International Conference on Machine Learning (ICML), volume 235 of Proceedings of Machine Learning Research, pp. 46412–46440. PMLR, 2024. URL https://proceedings.mlr.press/ v235/stanczuk24a.html. Piotr Tempczyk, Rafał Michaluk, Łukasz Garncarek, Przemysław Spurek, Jacek Tabor, and Adam Goliński. LIDL: Local intrinsic dimension estimation using approximate likelihood. In Proceedings of the 39th International Conference on Machine Learning (ICML), volume 162 of Proceedings of Machine Learning Research, pp. 21205–21231. PMLR, 2022. URL https://proceedings. mlr.press/v162/tempczyk22a.html. Constantin Udrişte. Convex Functions and Optimization Methods on Riemannian Manifolds, volume 297 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, 1994. doi: 10.1007/978-94-015-8390-9. Enrico Ventura, Beatrice Achilli, Gianluigi Silvestri, Carlo Lucibello, and Luca Ambrogioni. Manifolds, random matrices and spectral gaps: The geometric phases of generative diffusion. In International Conference on Learning Representations (ICLR), 2025. URL https: //arxiv.org/abs/2410.05898. Pascal Vincent. A connection between score matching and denoising autoencoders. Neural Computation, 23(7):1661–1674, 2011. doi: 10.1162/NECO a 00142. Xinghua Wang and Danfu Han. On dominating sequence method in the point estimate and Smale’s theorem. Scientia Sinica, Series A, 33:905–913, 1989. Layne T. Watson. Globally convergent homotopy methods: A tutorial. Applied Mathematics and Computation, 31:369–396, 1989. doi: 10.1016/0096-3003(89)90129-X. Li Kevin Wenliang and Ben Moran. Score-based generative models learn manifold-like structures with constrained mixing. In NeurIPS Workshop on Score-Based Methods, 2022. URL https: //arxiv.org/abs/2311.09952. arXiv:2311.09952. Edward Witten. Supersymmetry and Morse theory. Journal of Differential Geometry, 17(4):661–692, 1982. doi: 10.4310/jdg/1214437492. Manato Yaguchi, Kotaro Sakamoto, Ryosuke Sakamoto, Masato Tanabe, Masatomo Akagawa, Yusuke Hayashi, Masahiro Suzuki, and Yutaka Matsuo. The geometry of phase transitions in diffusion models: Tubular neighbourhoods and singularities. Transactions on Machine Learning Research, 2025. ISSN 2835-8856. URL https://openreview.net/forum?id=ahVFKFLYk2. Eric Yeats, Aaron Jacobson, Darryl Hannan, Yiran Jia, Timothy Doster, Henry Kvinge, and Scott Mahan. A connection between score matching and local intrinsic dimension. In NeurIPS Workshop on Structured Probabilistic Inference and Generative Modeling (SPIGM), 2025. URL https: //arxiv.org/abs/2510.12975.

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Preprint: Diffusion Inverse Step as a Strongly Convex Program

A

T HE C ALIBRATION S UITE

For a unit-radius manifold embedded at scale available in closed form: 1

2

−

r 2 +ᾱt 2 2σt

√

√ ᾱt , with z = r ᾱt /σt2 , the Gaussian convolution is 2

3

−

S ⊂ R : pt ∝ e 2 ρ +x2  ρ√ᾱ  3 +ᾱt − t 2 1 3 2σt I0 , S ⊂ R : pt ∝ e σt2

S ⊂ R : pt ∝ e

I0 (z);

r 2 +ᾱt 2 2σt

sinh z ; z

with ρ2 = x21 + x22 in the last case. Differentiating twice gives ε⋆ = −σt ∇ log pt and J ⋆ = −σt ∇2 log pt to machine precision; the Bessel ratio R = I1 /I0 is evaluated in exponentially scaled form and differentiated with the Riccati identity R′ = 1 − R/z − R2 . Schedule sweep, and the two normalisations. The contraction constant is ρ⋆g = |Bt |/σt+1 = 1 − e−ht , with the score evaluated at t+1; normalising by σt instead inflates it by exactly eht , and the difference decides the theorem. On the standard DDPM linear schedule (T =1000, β linear on [10−4 , 0.02]) the corrected constant reads 0.325699 at t=0, 0.218175 at t=1, 0.167878 at t=2, 0.118045 at t=4, 0.076633 at t=8, 0.027992 at t=30, 0.009941 at t=100, 0.005022 at t=300 and 0.010051 at t=998, with maximum 0.3256994484136135 attained at the first step — against 0.482989 for maxt |Bt |/σt under the old normalisation. The identity ρ⋆g = 1 − e−ht holds to a worst residual of 9.410 × 10−16 over six families × three strides, 9.402 × 10−16 over eleven schedule configurations, 3.192 × 10−16 over the whole point-cloud grid and 8.656 × 10−16 over the DDPM T =1000 grid; the constant 0.32569944841361353 is bit-identical across all three. The coarse schedules that broke the old reading do not break this one: T =100 gives 0.500687 where the old form gave 1.003, and T =100 with βend = 0.05 gives 0.623219 where the old form gave 1.654. Where the bound is attained. Table 2 sweeps seven families at five strides, and three rows carry the scope of Thm. 4. A cosine schedule has ᾱT = 0 exactly, so ρ⋆g → 1 at its last step and the printed margin is set by the library’s floor rather than by the schedule: we print the diffusers convention (β clipped at 0.999, giving 0.968377), while clipping ᾱ at 10−8 gives 0.935834 for the same nominal family, and neither number is a property of the cosine schedule. Rectified flow reads ρ⋆g = 1.000000 exactly at its top step, where ᾱ = 0 and ht = ∞: the bound is attained, not violated, and the exact statement is ρ⋆g < 1 for every schedule with ᾱT > 0, with equality only at a zero-terminal-SNR top step (Lipman et al., 2023). EDM/Karras spans 0.024484 at stride 1 to 0.681146 at stride 50, so the certificate’s margin varies by a factor 28 across families (Karras et al., 2022; Nichol & Dhariwal, 2021). Parametrisation is immaterial by algebra rather than by measurement: over N =400 matched −16 σ-grids max |hVE − hVP and max |ρ⋆,VE − ρ⋆,VP | = 2.9 × 10−16 while the meant t | = 8.9 × 10 g g value coefficient itself moves by 1.96 × 10−2 ; SD3/Flux static shifts m ∈ {1, 3, 6} and dynamic shifts µ ∈ {0.5, 1.0, 1.9} move ρ⋆g by at most 3.8 × 10−14 , because a translation of the log-SNR axis cannot change any ht .

B

F ULL P ROOFS

p √ Throughout, εϕ = ε⋆ , At = ᾱt+1 /ᾱt , Bt = σt+1 − At σt , λt = log( ᾱt /σt ) and ht = λt − λt+1 , strictly positive for any strictly decreasing ᾱ. Everything rests on one exact schedule identity, which involves no expansion in the step size: p
|Bt | = 1 − e−ht . At σt = σt ᾱt+1 /ᾱt = e−ht σt+1 , hence Bt = σt+1 1 − e−ht , ρ⋆g := σt+1 Since ht is a difference of log-SNRs, no reparametrisation of the noise axis and no shift of the log-SNR grid can change it; this is why the constant transfers unchanged between VP, VE and shifted flow-matching parametrisations (App. A). Proof of Theorem 1. By Tweedie–Miyasawa (3) at level t+1, ε⋆ (x, t+1) = −σt+1 ∇ log pt+1 (x), so F (x) = x − G(x) = x − At y − Bt ε⋆ (x, t+1) = x − At y + Bt σt+1 ∇ log pt+1 (x) = ∇Ψt (x) 16

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Table 2: Seven schedule families at five strides. ρ⋆g = 1 − e−ht is the Bayes contraction rate at the family’s worst step, wP = 1/(1 + ρ⋆g ) the Picard divergence boundary of Thm. 5, and the last column is the superseded |Bt |/σt normalisation, shown to make the size of the correction visible. Only a zero-terminal-SNR top step attains ρ⋆g = 1. ρ⋆g

1/ρ⋆g

wP

|Bt |/σt (old)

1 10 20 50 100

0.325699 0.786704 0.874046 0.943182 0.970781

3.070 1.271 1.144 1.060 1.030

0.7543 0.5597 0.5336 0.5146 0.5074

0.4830 3.6845 6.9180 16.3496 31.4357

cosine T =1000 (clip .999)

1 10 20 50 100

0.968377 0.996838 0.998419 0.999369 0.999686

1.033 1.003 1.002 1.001 1.000

0.5080 0.5008 0.5004 0.5002 0.5001

0.9684 3.1796 5.7595 13.1621 25.2536

sigmoid T =1000

1 10 20 50 100

0.293621 0.701663 0.786489 0.868084 0.913396

3.406 1.425 1.271 1.152 1.095

0.7730 0.5877 0.5598 0.5353 0.5226

0.4156 2.3501 3.6779 6.5530 10.4442

scaled-linear SD T =1000

1 10 20 50 100

0.294020 0.703257 0.788630 0.870703 0.915195

3.401 1.422 1.268 1.148 1.093

0.7728 0.5871 0.5591 0.5346 0.5221

0.4163 2.3596 3.6976 6.5718 10.2094

EDM/Karras N =1000 (VP)

1 10 20 50 100

0.024484 0.216531 0.381110 0.681146 0.880607

40.843 4.618 2.624 1.468 1.136

0.9761 0.8220 0.7241 0.5948 0.5317

0.0251 0.2764 0.6158 2.1362 7.3747

VE/Karras N =1000

1 10 20 50 100

0.024484 0.216531 0.381110 0.681146 0.880607

40.843 4.618 2.624 1.468 1.136

0.9761 0.8220 0.7241 0.5948 0.5317

0.0251 0.2764 0.6158 2.1362 7.3757

FlowMatchEuler N =1000

1 10 20 50 100

1.000000 1.000000 1.000000 1.000000 1.000000

1.000 1.000 1.000 1.000 1.000

0.5000 0.5000 0.5000 0.5000 0.5000

1.0010 10.0100 20.0200 50.0501 100.1001

family

stride k

DDPM-linear T =1000

with Ψt (x) = 12 ∥x−At y∥2 +Bt σt+1 log pt+1 (x); the residual is a gradient field for every data law, because it is an affine function of a score. Differentiating once more and using J ⋆ = −σt+1 ∇2 log pt+1 , Bt  Ct+1 (x)  Ct+1 (x) ∇2 Ψt (x) = I − Bt J ⋆ (x, t+1) = I − I− = e−ht I + ρ⋆g , 2 2 σt+1 σt+1 σt+1 by (7) at t+1 and 1 − ρ⋆g = e−ht . Since Ct+1 ⪰ 0 for every p0 , ∇2 Ψt ⪰ e−ht I, so Ψt is strongly convex with modulus exactly µ = e−ht and µ + ρ⋆g = 1. Picard iteration x 7→ G(x) = x − F (x) = x − ∇Ψt (x) is therefore gradient descent on Ψt with step size exactly 1 — not approximately, and with no freedom to choose otherwise. Inside the noise tube of a uniform M the Fermi values of the 2 posterior covariance are c⊥ = 0 on the D − d normal directions and c∥ = σt+1 /(1 − rκi ) on the d 2 (d) −ht (D−d) tangential ones, so at r = 0 the spectrum of ∇ Ψt is {1 , (e ) }: its condition number is eht , independent of the data law, of d, of D and of curvature, and smin (I − Bt J ⋆ ) = e−ht exactly. Convexity of − log pt+1 is not assumed anywhere; what is used is that log pt+1 is the log-partition function of the denoising exponential family, whose Hessian is a covariance. 17

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Figure 2: Bayes-limit calibration on analytically integrable manifolds; no trained model is involved. (a) The two dimensionless certificates St (solid) and σt2 λ̂ (dashed). The offset of St from 1 is not error: it is the closed-form curvature deficit of (8), matched to all printed digits in codimension one, with an O(σt4 ) residual. (b) The scaling exponents (14) converge to +1 (normal) and −1 (tangential); the value 0 often assumed for the tangential family is the dotted threshold, not a limit. (c) The same functional evaluated at the sampled point returns the ambient D at every noise level (Prop. 6) and at the noiseless point returns the intrinsic d (Prop. 7); error bars are 95% intervals over 2 × 104 draws.

Figure 3: The descent-based Łojasiewicz estimator is uninformative by construction (Remark 11). (a) For a quadratic energy the log–log locus is the line of slope 12 over ten decades whether the iteration converges or diverges; the two traces coincide with the theorem line. (b) What the estimator returns on analytic energies of known exponent: 12 on quadratic and Morse–Bott energies including under a divergent step, a different value under a stable step on the same energy, and 0.077 on a quartic energy whose true exponent is 34 (dotted arrow to the truth). Open diamonds: the field estimator of (13) on the same energies, which returns the true exponent in all four cases. Proof of Proposition 3. At Bayes J ⋆ = −σt ∇2 log pt is symmetric, so sym J ⋆ = J ⋆ and by (7) σt λmax (J ⋆ ) = 1 − λmin (Ct )/σt2 ≤ 1, with equality iff Ct is singular in the extremal direction. No manifold, reach, unimodality or tube hypothesis enters, and the statement is pointwise in x and uniform in t. Two consequences are used in §5: σt λmax (sym J) > 1 falsifies Bayes optimality outright, and σt tr J ≤ D for the same reason, since σt tr J ⋆ = D − tr(Ct )/σt2 . Proof of Theorem 2. By Theorem 1, ∇2 Ψt ⪰ e−ht I ≻ 0 for every x, so Ψt is strictly convex and F = ∇Ψt has exactly one zero. Replacing ε⋆ by a trained εϕ destroys the gradient structure but not the argument:
Fϕ (x) = x − At y − Bt εϕ (x, t+1) satisfies ⟨Fϕ (x) − Fϕ (z), x − z⟩ ≥ 1 − Bt λmax (sym Jϕ ) ∥x − z∥2 along the segment, so Fϕ is strictly monotone — hence injective — unless σt+1 λmax (sym Jϕ (ξ, t+1)) > 1/(1 − e−ht ) for some ξ. The condition is therefore necessary for a second solution and, being a bound on the symmetric part, not sufficient. At the threshold the fold is the only bifurcation available: a multiplier +1 of DG = Bt Jϕ is exactly the displayed inequality, a multiplier −1 produces period doubling but no second root, and a complex pair is excluded whenever DG is symmetric, which it is at Bayes and approximately so wherever the skew index is small. In the two-atom model the new roots sit at x = ρ⋆ c/(ρ⋆ c − 1), where the posterior covariance vanishes and |G′ | → ρ⋆ c — the threshold quantity itself — so they are repelling exactly 18

Preprint: Diffusion Inverse Step as a Strongly Convex Program

when they exist. Multiplicity is thus a Newton observable and is invisible to Picard, which is why we report it through the signed pair and not through a failed iteration. Proof of Theorem 5. Combine Theorem 1 with the Fermi values of Ct+1 . The Picard multipliers are
2 the eigenvalues of I − ∇2 Ψt = ρ⋆g I − Ct+1 /σt+1 , real because ∇2 Ψt is symmetric. Displacing inward along a principal normal by r and writing wi = rκi , the exact tube values c⊥ = 0 and 2 c∥,i = σt+1 /(1 − wi ) give the multiplier set {+ρ⋆g (× D−d)} ∪ {−ρ⋆g wi /(1 − wi ) (× d)} for every σt and every data law. (1) The dominant multiplier changes sign when w/(1 − w) = 1, i.e. wosc = 21 ; 2 the same condition is λmax (Ct+1 ) = 2σt+1 , which is exactly the failure of Assumption 2, so the oscillation boundary and the unimodality edge are one object and neither depends on the schedule. (2) The dominant multiplier reaches magnitude 1 when ρ⋆g w/(1 − w) = 1, i.e. wdiv = 1/(1 + ρ⋆g ); equivalently λmax (∇2 Ψt ) = e−ht + ρ⋆g /(1 − w) = 2, so the Picard divergence boundary and the damping threshold η < 2/Λt are the same equation. Beyond the statement, differentiating λmax (∇2 Ψt ) in r gives the ray-Lipschitz constant LH (w) = ρ⋆g κmax /(1−w)2 , while ∥(∇2 Ψt )−1 ∥ ≤ eht everywhere by Theorem 1, so the Newton–Kantorovich product is aLH = c κmax /(1 − w)2 with c = eht − 1: free of the data law, of the density and of the dimension, and divergent as w → 1, so no certified ball reaches the focal point. (3) Carrying the next order of the Fermi √ determinant, the correction to wdiv is quadratic in σ̃ = σ/ ᾱ with coefficient the trace of the squared second fundamental form, and the correction to wosc is quartic; the shape constant is fixed by measurement rather than by this sketch, and §5.1 reports it at 1.00007–1.00178 on the closedform targets and 1.008–1.020 on real clouds. Only ∥II∥2 reproduces both the codimension and the dimension dependence: on a curve ∥II∥2 = κ2 and the two candidate normalisers coincide, which is why the 27-setting scan over S 1 ⊂ R2 , S 2 ⊂ R3 and S 1 ⊂ R3 is the only place in this study that can separate them. Proof of Theorem 4. Bt = σt+1 (1 − e−ht ) is exact, so ρ⋆g = 1 − e−ht < 1 for every schedule with ht > 0, i.e. for every strictly decreasing ᾱ with ᾱT > 0; the earlier closed form was an expansion of a quantity that needs none. The Lipschitz factorisation is definitional, ρg = |Bt | supξ s1 (J(ξ, t+1)) = ρ⋆g St+1 , and St+1 ≤ 1 under Assumption 2 by (7). The damping statement follows from Theorem 1: gradient descent on an e−ht -strongly-convex, Λt -smooth Ψt converges for every η < 2/Λt , optimally at η ⋆ = 2/(e−ht + Λt ) with rate (Λt /µ − 1)/(Λt /µ + 1) in the condition number Λt /µ; inside the tube at r = 0 that number is eht and the rate is tanh(ht /2). Finally µ + Λ( 12 ) = 2 identically, so unit-step Picard is optimally damped precisely at the oscillation shell and nowhere else. Three levels, and why only one of them carries geometry. The three statements above separate objects that the literature treats as one. The solution set is trivial at the Bayes limit — one root, by strict convexity, for every data law and every schedule — so a second solution is a hypothesis-free certificate of model error and carries no information about M. The solver orbit can be pathological at the Bayes limit and certifies nothing about the model: on an exact two-atom Bayes score with a unique minimiser and ∇2 Ψt ⪰ 0.4I everywhere, λmax (∇2 Ψt ) = 2.011, 3.022, 4.033, 7.067 at ρ⋆g = 0.1, 0.2, 0.3, 0.6, and undamped Picard reaches tolerance from 1 of 2001 starts on [−6, 6] — the one being the fixed point itself, so 0 of 2000 non-trivial starts — while it is captured by an attracting 2-cycle at ±0.374819; at η = 1/Λt the same sweep converges from 100.00% of starts. What does carry geometry is the convergence domain: its two boundaries are the closed forms of Theorem 5, and they are curvatures. A binary convergence readout is the wrong instrument for the first two levels and the right one only for the third, and even there its rate vanishes at Λt = 2, so the sharp object is the theorem and not the iteration count. Numerical verification of Thm. 1. Two atoms at ±1, σ = 0.3, 200 001 grid points: the Hessian identity holds to 8.9 × 10−16 , 1.8 × 10−15 , 1.8 × 10−15 and 3.6 × 10−15 at ρ⋆g = 0.2, 0.4, 0.6, 0.8; µ = e−ht to 2.8 × 10−17 ; µ + ρ⋆g = 1.000000 on 4 of 4 rows; and F = ∇Ψt to the central-difference floor, ≤ 4.5 × 10−11 at dx = 10−5 , with clean O(dx2 ) behaviour from 10−2 down to 10−5 . On the Bessel circle at σ = 0.01 the two eigenvalues are 0.99998 and 0.67432 against e−ht = 0.67430. The first Kantorovich constant is attained: a/eht = 0.9947 over ten exact-score settings, range 0.98751–0.99932. 19

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Numerical verification of Thm. 2. The threshold is verified in an analytic model only, never on a trained score. At Bayes, σt+1 λmax (J ⋆ ) = 1.000000 and there is exactly one root at ρ⋆g = 0.05, 0.20, 0.50, 0.90, 0.99 with min ∇2 Ψt = 0.95, 0.80, 0.50, 0.10, 0.010. With an amplitudescaled score εϕ = γ ε⋆ at ρ⋆g = 0.6 the root count is 1 at ρ⋆g γ = 0.600, 0.840, 0.960, 0.996 and 3 at 1.050, 1.200, 1.800, bracketing the predicted 1; the new roots sit at ±21.00, ±6.00, ±2.25, i.e. at x = ρ⋆g γ/(ρ⋆g γ − 1) exactly, with multipliers ρ⋆g γ = 1.0500, 1.2000, 1.8000 — so they are repelling iff they exist. Two limits of that experiment must be printed with it: the root search spans |x| ≤ 60, so onset is undetectable for γ ≤ 1.695 and the bracket γ ∈ (1.66, 1.75] cannot be tightened in this configuration; and the planned pitchfork-on-a-network section was not built, so the certificate has zero observations on any trained score. Numerical verification of Thm. 5. On the Bessel-exact circle, wosc = 0.4999999599, 0.4999999975, 0.4999999998 at σ = 0.02, 0.01, 0.005, identical at ρ⋆g = 0.278 and 0.6527, with residual ratios 16.03 and 16.26, i.e. O(σ 4 ), while the wdiv residual falls with ratios 4.005 and 4.001, i.e. O(σ 2 ), and the coefficient of part (iii) is measured at 1.00178 falling to 1.00007. The ∥II∥2 normalisation, and not κ2max , is what the data selects: over a 27-setting scan on S 1 ⊂ R2 , S 2 ⊂ R3 and S 1 ⊂ R3 at three noise levels and three schedules, the two-term form reproduces every measured wdiv at the scan’s own resolution, 27 of 27, where the one-term form misses 8 of 27; S 1 ⊂ R3 is identical to S 1 ⊂ R2 in 9 of 9 settings, so codimension has no effect, and S 2 sits one grid step high in exactly the two settings the doubled coefficient predicts.

C

D EFERRED M ETHOD M ATERIAL

This section carries the proofs and the attributions of §4.5; the statements themselves are in the main text. The Fokker–Planck readout at the noiseless point. ˆ = E[∥ε∥2 | xt =x] pointwise, and Proposition 7 (The noiseless-point readout). At Bayes d(x)  2  2 ˆ ◦ ) = d + σt ∥II∥2 − 1 ∥H∥2 + σt ∆M ρ0 + O(σ 4 ). d(x t 2 2ᾱt ᾱt ρ0
The residual is ε⋆ (x◦ ) = − √σᾱt t ∇M log ρ0 + 12 H + O(σt2 ), which is − 2√σtᾱt H + O(σt2 ) for a uniform ρ0 , but the trace term contributes at the same order, so the total correction is signed: +σt2 κ2 /(4ᾱt ) for a curve of curvature κ, exactly 0 for the round S 2 , and − d(d−2) σt2 /ᾱt for S d ⊂ 4 d+1 R , negative for d ≥ 3. Proof. From the Fermi expansion of §4.1, log pt = −r2 /2σt2 − 12 log det(I − rAν ) + log ρ0 + const. Differentiating and applying (3) gives a normal component r/σt − σ2t tr Aν + O(σt r), and tr Aν contracted with ν is H; at r = 0 only that term survives. The trace term contributes D − d at leading order by (6). Proof of Prop. 6. Integration by parts — the implicit score-matching identity (Hyvärinen, 2005) ˆ = D + E⟨εϕ , εϕ − ε⋆ ⟩ = D − δt /2 by — gives Ept [−σt tr J] = −E⟨εϕ , ε⋆ ⟩ = −E⟨εϕ , ε⟩, so E[d] R (11). At Bayes dˆ = D + σt2 ∆pt /pt pointwise and RD ∆pt = 0 by the divergence theorem. Two consequences of the vacuity. Averaged over sampled points the readout is biased upward from D by exactly γ(γ−1)Π⋆t + E∥e∥2 (Prop. 8), and Ept [−σt tr J] = −E⟨εϕ , ε⟩ makes Hutchinson redundant in expectation there, one dot product with the injected noise replacing 32 probes. None of this is a criticism of the published protocol but the reason for its design: Kamkari et al. (2024, Eq. 13) evaluate at the scaled clean point ψ(t)x = x◦ , and Prop. 6 says that choice is forced. Pointwise the ˆ t ) = D + σ 2 ∆pt /pt is not constant; the obstruction is to the average, statement is weaker, since d(x t and §5.2 supplies a second one at the same point. 20

Preprint: Diffusion Inverse Step as a Strongly Convex Program

The DSM readout. Lt is the denoising-score-matching objective in its noise parametrisation (Vincent, 2011). Proposition 8. The two readouts Lt /ᾱt and (D − Πt )/ᾱt differ exactly by δt /ᾱt and cannot corroborate each other where δt ̸= 0. Writing εϕ = γε⋆ + e with E⟨ε⋆ , e⟩ = 0, the decomposition is exact: δt = −2γ(γ − 1)Π⋆t − 2E∥e∥2 . The rank ceiling. Proposition 9. Local PCA on K Tweedie-projected samples satisfies dˆ ≤ K − 1; centring is intrinsic to that estimator. A score matrix built from K score evaluations caps the estimated codimension at \ D − d ≤ min(D, K), and at K − 1 if the evaluations are centred first, hence dˆ ≥ D − K + 1 in the centred case. Proof. The estimated subspace is the span of the K evaluations — the tangent space for local PCA, the normal space for the score matrix — so its dimension is at most K; centring makes the K columns sum to zero and removes one further degree of freedom. Pinned at the ceiling, local PCA reports a lower bound on d and the score-rank estimator an upper bound; neither binds at the prescribed K = 4D. Morse–Bott, not Morse. √ Remark 10. Et does not vanish on ᾱt M: by Prop. 7, Et (x◦ ) = σt2 ∥H∥2 /(8ᾱt ) + O(σt3 ), and the √ σ2 zero set of ε⋆ is the curvature-shifted manifold M⋆t = { ᾱt x0 + 2t H(x0 ) + O(σt3 )}. Along M⋆t the Hessian has a d-dimensional kernel, so Et is Morse–Bott and never Morse, and any derivation assuming a positive-definite Hessian is vacuous here. Transverse to M⋆t it is (I − P )/σt2 + O(1), whose D − d eigenvalues are equal to leading order — exactly equal for the limiting energy 12 dist2 , diagonal along a normal ray in one offset-independent frame with all normal eigenvalues exactly 1 (Ambrosio & Mantegazza, 1998), splitting only at relative order σt2 . It is this transverse nondegeneracy — automatic in the noise tube and unaffected by that splitting — that delivers the exponent of (13). That Morse–Bott forces the optimal exponent is known abstractly (Feehan, 2017; Rebjock & Boumal, 2025); (13) adds the constant, and (12) turns the intercept of a log–log fit into a dimensionless stiffness with Bayes value 1. The Fokker–Planck functional In DDPM parametrisation the functional of Kamkari et al. (2024) is ˆ = D − σt tr J(x, t) + ∥εϕ (x, t)∥2 , d(x) (10) |{z} | {z } | {z } ambient

removes the codim. iff εϕ near-projects

curvature term, = (σt ∥H∥/2)2 at Bayes

with tr J by Hutchinson probes (Hutchinson, 1989). At Bayes dˆ = D + σt2 ∆pt /pt , the form that makes Props. 7 and 6 immediate. The optimality gap

With shared noise draws and E∥ε∥2 = D, expanding (2) gives exactly δt = D − Πt − Lt = 2E⟨εϕ , ε − εϕ ⟩ = 2E⟨εϕ , ε⋆ − εϕ ⟩,

(11)

the last step by the tower property, so δt is the orthogonality defect of εϕ against the L2 projection ε⋆ . The dimensionless stiffness read off the Hamilton–Jacobi intercept is σt2 λ̂ := σt2 · 21 e2ĉ −→ 1.

(12)

Rank readouts Pinned at the ceiling of Prop. 9, local PCA reports a lower bound on d and the score-rank estimator an upper bound; neither binds at K = 4D. The largest-difference rule is maximised at the top edge whenever s1 detaches from the bulk; the ratio rule is maximised at the bottom edge whenever smin → 0, automatic at K ≤ D by Prop. 9 but not only there. The pure-noise null bounds false gaps and leaves power the separate question H5 settles. 21

Preprint: Diffusion Inverse Step as a Strongly Convex Program

The score energy as an action. At Bayes Et (x) = r2 /(2σt2 ) + O(σt ) is, to leading order, the squared distance to the manifold rescaled by the noise — the classical action of the diffusion at time σt2 (see Calin, 2024, §6.5.5, for this identification in the Riemannian setting), and inside the reach the nearest-point projection is unique with ∥∇r∥ = 1, the eiconal property (Federer, 1959). The transverse law and the scaling exponents (§4.5). The last identity is transverse. Inside the reach r is C 1 with ∥∇r∥ = 1 (Federer, 1959) and Et = r2 /(2σt2 ) + O(σt ), so with only transverse nondegeneracy √ √ r 2Et 2 1 ∥∇Et ∥ = 2 = =⇒ θ = 2 , cL = (13) σt σt σt on the whole critical manifold, and with its constant. Et is Morse–Bott there and never Morse, so any derivation assuming a positive-definite Hessian is vacuous (Rem. 10), and the intercept of a log–log fit gives a dimensionless stiffness σt2 λ̂ → 1. Remark 11 (Why a descent cannot see this). For E = 12 λr2 under any linear iteration rk+1 = (1 − ηλ)rk , log ∥∇Ek ∥ = 12 log Ek + 12 log(2λ) identically, converging or diverging. A descent estimator of θ therefore returns 12 with R2 = 1 whenever a single scale governs the active spectrum — automatic here, the transverse eigenvalues coinciding — and otherwise reports the step size and the iteration budget. Equation (13) must be read from the field along a normal ray. Finally, by (6) the normal singular values of J ⋆ (x◦ ) scale as σt−1 and the tangential ones as σt+1 , not σt0 , so matching singular values across two levels σA < σB gives exponents
log sj (σA )/sj (σB ) −→ ±1 (normal/tangential), (14) νj := log(σB /σA ) separated by 2, with natural threshold 0 and a classifier invariant to any state- and noise-independent amplitude error. Two conditions bind it: a random m-dimensional probe subspace meets the tangent space only if m > D − nd, so ν must be read on a full basis, a random m ≪ D probe pinning every exponent at +1 by construction; and at a free boundary the outward tangential direction is truncated and reads as normal, where the exact score returns 174 normal directions against a true codimension of 128. We report ν as family medians, never as counts. Why the stiffness deficit is measurable only in low codimension. S⋆t of (8) vanishes iff some normal direction has vanishing shape operator, which is automatic in high codimension: minν ∥Aν ∥2F = λmin (Q) for Qab = ⟨IIa , IIb ⟩, and IIa ∈ Sym(d) forces rank Q ≤ d(d + 1)/2, so S⋆t = 1 exactly whenever d(d + 1)/2 < D − d — at D=3072, for every d ≤ 76. Wenliang & Moran (2022) already read a trained score Jacobian’s spectrum against this scale at a fixed tolerance; (8) is the closed form of what that tolerance stands in for.

D

D EFERRED D ERIVATIONS

Exact DSM readout on a linear manifold. With M = range(U ), U ∈ RD×d orthonormal, and x0 = U z, z ∼ N (0, Id ): Σt = ᾱt U U ⊤ + σt2 I and ε⋆ (x) = σt Σ−1 t x. Each normal direction contributes 0 to the loss (ε⋆⊥ = ε⊥ exactly) and each tangential direction contributes E((1 − σt2 )ε∥ − √ σt ᾱt z)2 = ᾱt2 + σt2 ᾱt = ᾱt . Hence L⋆t = ᾱt d exactly at every t, so Lt /ᾱt = d on this model. ˆ ◦ ) = ᾱt (d + σ 2 ∥z∥2 ), so at the typical radius the schedule-corrected The same computation gives d(x t ◦ ˆ )/[ᾱt (2 − ᾱt )], which is the schedule-corrected normalisation used throughout form of (10) is d(x §5. Amplitude decomposition. With εϕ = γε⋆ + e, the amplitude multiplier of §4.2, and E⟨ε⋆ , e⟩ = 0: Πt = γ 2 Π⋆t + E∥e∥2 and E⟨εϕ , ε⟩ = γΠ⋆t , so δt = −2γ(γ − 1)Π⋆t − 2E∥e∥2 exactly. Both terms are non-positive for γ > 1, which is why δt screens but does not test. Descent tautology. For E = 12 λr2 and rk+1 = (1 − ηλ)rk : Ek = 12 λrk2 and ∥∇Ek ∥ = λrk , so log ∥∇Ek ∥ = 21 log Ek + 12 log(2λ) for every k, independently of η and of whether |1 − ηλ| < 1. The fit is exact, so R2 = 1 by construction. In general, with qk := ∥∇Ek ∥2 /(2Ek ) ∈ [λmin , λmax ], 22

Preprint: Diffusion Inverse Step as a Strongly Convex Program

log ∥∇Ek ∥ = 12 log Ek + 12 log(2qk ) identically for every spectrum and step size, so a log–log regression returns 12 + ∆ log q/(2∆ log E) and deviates from 12 by at most log κ over twice the fitted range of log E. The published hyper-parameters imply ηλmax ≈ 166, i.e. 83× the stability threshold 2/λmax , and across ηλmax from 0.5 to 166 the readout stays in [0.4974, 0.5009] with R2 ≥ 0.9998. P 1 The same cancellation holds verbatim for E = 2 i λi ri2 whenever the modes with ri ̸= 0 share one λ — the transversally isotropic case of Remark 10, where the intercept returns the single transverse eigenvalue 1/σt2 at any step size. With a spread transverse spectrum the log–log locus is governed asymptotically by the slowest-decaying mode, so the intercept returns λj with j = arg maxi |1−ηλi |: the largest eigenvalue under a divergent step and the smallest under a stable one. Equation (12) is read from the field along a normal ray, where no step size enters. Assumption 2 as geometry. In Fermi coordinates at second order about a uniform M, the posterior at offset r along the normal ν has λmax (Ct ) = σt2 /(1 − rκmax ) + O(σt4 ), so λmax (Ct ) ≤ 2σt2 iff r ≤ 1/(2κmax ): half the focal radius, which coincides with half the reach only when the reach is curvature-attained. Quadrature on the uniform circle gives λmax (Ct )/σt2 = 1.3330, 1.6661, 1.9992, 2.4987, 3.9968 at r = 0.25, 0.40, 0.50, 0.60, 0.75 against the predicted 4/3, 5/3, 2, 5/2, 4. Near a bottleneck the exact value saturates at (ℓ/2)2 rather than diverging, so the assumption fails only inside a slab of half-width O(σt2 ) about the medial sheet. Separately, log√ concavity of the law of ᾱt x0 passes to the posterior with ∇2 (− log) ⪰ σt−2 I, so Brascamp–Lieb 2 gives Ct ⪯ σt I and margin 2. A lower edge for curvature readouts. The Bayes residual at the noiseless point is (r/σt )ν − 2 (σt /2)H, so the √ two terms cross at r = σt ∥H∥/2: any curvature readout taken from the residual needs dist(x, ᾱt M) ≪ σt2 ∥H∥/2. At the CIFAR probe (σt = 0.0264) that is r < 3.5 × 10−4 against a data radius of 55.4, a relative positional accuracy of 6 × 10−6 . This is the lower-side counterpart of (15) and the reason we report ∥ε⋆ (x◦ )∥ only as a ratio against a known H on integrable manifolds, and never invert it on a trained model. Reach of a product, and the factorisedP window. For a closed product M = M1 ×· · ·×Mn in the product ambient space, dist(x, M)2 = p dist(xp , Mp )2 , so the argmin separates and x has a nonS unique nearest point iff some xp does. Hence Med(M) = p πp−1 (Med(Mp )), and the medial-axis definition of the reach (Federer, 1959; Aamari et al., 2019) gives reach(M) = minp reach(Mp ); the Clifford torus S 1 × S 1 ⊂ R4 checks this, with focal radius and half-bottleneck both 1. The normal exponential map is therefore a diffeomorphism on the polydisc {dist(xp , Mp ) < reach(Mp ) ∀p}, which strictly contains the metric tube of radius minp reach(Mp ). Because εϕ acts per point, the noised law, the score, the Jacobian and every readout factorise, so a sampled xt is in the regime of validity when maxp σt ∥ξp⊥ ∥ < reach(Mp ) — and E[maxp χk ] is 2.83 (k=1) and 3.28 (k=2) at √ n=128, with 95th percentiles 3.54 and 3.96, against D − d = 11.3 and 16.0. For the two additive readouts the conclusion is stronger: both are sums over factors, so their relative bias equals the singlefactor relative bias and carries no n-dependence at all. This criterion is necessary, not sufficient: √ Niyogi et al. (2011, Thm. 1) require σ D − d < 0.0067 reach(M), two orders of magnitude stricter. The reach audit, and the three window criteria side by side. Reaches are computed after normalisation, as the smaller of the focal radius 1/κmax and half the shortest bottleneck, and the cause is read off which of the two binds. Two of the four values used in an earlier draft were not reaches: 0.36 is the two-sphere focal radius and 0.43 matches neither the trefoil’s reach nor its focal radius in this run. Corrected: sphere 0.9971 (curvature), torus 0.2587 (curvature), two spheres 0.2825 (bottleneck), swiss roll 0.2090 (bottleneck), trefoil 0.2021 (bottleneck), reproduced to ≤ 3 × 10−10 on the curvature side. At the smallest probed noise (σt = 0.04256) the three criteria read M sphere torus two spheres swiss roll trefoil

reach(M)

σt / reach (at x◦ )

√ σt D − d/ reach (15)

polydisc criterion

0.9971 0.2587 0.2825 0.2090 0.2021

0.043 0.165 0.151 0.204 0.211

0.483 1.861 1.704 2.303 3.369

0.151 0.582 0.533 0.721 0.834

23

Preprint: Diffusion Inverse Step as a Strongly Convex Program

so all five classes are inside the product window at that level, where the metric-tube reading of (15) would exclude four of them. The effect of the correction is to remove an excuse rather than to license a claim: the trefoil reads 2.99× its flattened target while inside the corrected window, so its failure to separate d=1 from d=2 moves from the protocol column to the structural one. The Fermi window against the training support, and why the two cannot both hold. Converting a convergence-shell radius into a curvature uses the second-order Fermi expansion, whose validity we require at σt+k κmax ≤ 0.1; the probe must also stay where the model has training mass, r ≲ 1.4 σt+k . Since the shells sit at wosc = 21 and wdiv = 1/(1 + ρ⋆g ) and w = rκmax , the two conditions together bound the observable depth by w ≤ 1.4 σt+k κmax ≤ 0.14, short of the oscillation shell by 3.6× and of the divergence shell by 5.6×. Neither bound refers to the model, the manifold, the schedule or the training budget, so the conflict is structural. Table 3 shows it setting by setting: the Fermi condition holds in 1 of 25, the looser r ≤ 2σt+k support criterion tabulated there in 18 of 25, and both in 0 of 25; σt+k κmax spans 0.059 to 3.712 against a 0.10 window. This is the reason every trained shell in §5 is measured against the exact score at matched noise and never against the closed form, and the reason the oracle-minus-closed-form gap is reported as the predicted 12 σ̃ 2 ∥II∥2 correction rather than as an error. One convention must be named: the σκ column is built at t+k, where the network is queried; a companion pipeline tabulates the same quantity at t, which is smaller by a factor 1.376 on every row. Table 3: The diagnostic grid. Fermi: σt+k κmax ≤ 0.10. Support: the predicted oscillation shell within 2σt+k of the base point. Radii are in units of σt+k . setting

M

t=2, k=1

σt+k κmax

Fermi

rosc /σ

rdiv /σ

support

both

sphere torus swiss roll trefoil two spheres

0.0588 0.2265 0.1887 0.1167 0.1628

yes no no no no

8.51 2.21 2.65 4.28 3.07

13.37 3.47 4.16 6.73 4.82

no no no no no

no no no no no

t=2, k=5

sphere torus swiss roll trefoil two spheres

0.1230 0.4742 0.3952 0.2444 0.3409

no no no no no

4.06 1.05 1.27 2.05 1.47

4.92 1.28 1.53 2.48 1.77

no yes yes no yes

no no no no no

t=2, k=20

sphere torus swiss roll trefoil two spheres

0.3742 1.4422 1.2019 0.7433 1.0370

no no no no no

1.34 0.35 0.42 0.67 0.48

1.42 0.37 0.44 0.71 0.51

yes yes yes yes yes

no no no no no

t=10, k=20

sphere torus swiss roll trefoil two spheres

0.5207 2.0072 1.6728 1.0344 1.4432

no no no no no

0.96 0.25 0.30 0.48 0.35

1.15 0.30 0.36 0.58 0.41

yes yes yes yes yes

no no no no no

t=30, k=20

sphere torus swiss roll trefoil two spheres

0.9630 3.7118 3.0933 1.9129 2.6688

no no no no no

0.52 0.13 0.16 0.26 0.19

0.71 0.18 0.22 0.36 0.26

yes yes yes yes yes

no no no no no

Rank-readout budget, at the prescribed K. Stanczuk et al. (2024) prescribe K = 4D score evaluations (4d in their notation, where d is ambient: their Alg. 1 sets d ← dim x0 ), i.e. 1536 at D=384 and 12288 at D=3072, and this run is at protocol. Their derivation makes the normal block of the score matrix a (D − d) × K √ Gaussian√matrix, whose Marchenko–Pastur aspect γ = (D − d)/K fans the singular values by (1 + γ)/(1 − γ) before any signal is present; K = 4D is precisely the budget that pins γ ≤ 1/4, at a fan-out of 3.0. Our pure-noise nulls at that budget are correspondingly tight: largest consecutive gap ratio 1.0219 mean and 1.0325 max at (K, D) = (1536, 384) over 25 replicates, and 1.0054 mean, 1.0060 at the 95th percentile and 1.0061 max at (12288, 3072) over 5, 24

Preprint: Diffusion Inverse Step as a Strongly Convex Program

against an abstention threshold of 2.01. Raising K to protocol retired our own positive reading as well as the negative it was meant to rescue. On the exact score the ratio rule now recovers the true codimension on four of five classes (index 128, 128, 256, 128 at gap ratios 12.31, 3.39, 5.70, 4.33) and the difference rule on five of five; on every trained setting both rules fail, the ratio rule abstaining at rmax = 1.023–1.579 and the unthresholded difference rule emitting a confident dˆ ≈ D − 2 (381, 383, 383, 383, 39 converged and 1, 383, 383, 383, 383 undertrained). Two protocol deviations remain, and are ours rather than theirs: we centre the score matrix and Stanczuk et al. do not, and we threshold a ratio at 2.01 and abstain below it where their rule is unthresholded. Neither rule dominates. The abstention is not free: on the swiss roll — the one class with a boundary — the oracle ratio rule fires at rmax = 2.832 at index 381 and returns dˆ = 3 instead of 256, a confident wrong answer at the prescribed budget with no structural zero to blame, while the difference rule returns 128 → 256 correctly on the same spectrum. We therefore report both rules everywhere and claim neither.

E

F ROM THE S MALE S URROGATE TO THE K ANTOROVICH T RIPLE

An earlier version of this appendix reported Smale’s α-theory (Smale, 1986) for F (x) = x − G(x), with α √ = βS γS adjudicated against the three constants in √circulation — 0.130707 (Smale, 1986), (13 − 3 17)/4 ≈ 0.157671 (Blum et al., 1998) and 3 − 2 2 ≈ 0.171573 (Wang & Han, 1989). We withdraw that reading, for three reasons, and replace it with a certificate every constant of which is measurable. Why the surrogate does not hold. The surrogate bounded γS = supk≥2 ∥DF −1 Dk F/k!∥1/(k−1) by its k=2 term. That step is a modelling assumption, and it is false at the Bayes limit on a curved manifold: the sequence γk is increasing over the range we can evaluate and does not attain its supremum at k = 2, so α ≤ α̂/2 is not an upper bound and no verdict follows from it. More fundamentally, γS is a supremum over all higher posterior cumulants of the denoising family; it has no closed form even at Bayes, and it is infinite for any piecewise-linear block, so it is not a quantity a probe can report. Finally, the Neumann bound ∥DF −1 ∥ ≤ (1 − ρg )−1 that the surrogate used needs ρg < 1 and hence Assumption 2, whereas Theorem 1 gives the sharper bound unconditionally: DF = ∇2 Ψt is symmetric with λmin ≥ e−ht , so ∥DF −1 ∥ ≤ eht with no hypothesis at all, and by Eckart–Young (Eckart & Young, 1936) smin (I − Bt J ⋆ ) = e−ht is the distance from the inversion operator to ill-posedness. Measured, that bound is an equality rather than a bound with slack: a/eht = 0.9947 on average over ten exact-score settings, range 0.98751–0.99932. The replacement. Kantorovich’s theorem (Kantorovich & Akilov, 1982; Ferreira & Svaiter, 2002) needs only first derivatives and a local Lipschitz constant, all three of which we can measure and each of which has a Bayes reference from Theorems 1 and 5: a = ∥(∇2 Ψt )−1 ∥ against eht , 2 b = ∥(∇2 Ψt )−1 F ∥ against c σt+1 ∥H∥/2, and L = Lip(∇2 Ψt ) against ρ⋆g κmax . The certificate is ℓ := abL ≤ 12 . It holds in 20 of 20 point-cloud settings (4.64 × 10−3 –8.54 × 10−2 on the exact score, 8.89 × 10−4 –2.83 × 10−3 on the trained one) and in 4 of 4 CIFAR-10 settings (1.997 × 10−4 – 4.415 × 10−3 ); Table 4 gives every entry against its reference. For a non-analytic F the literature’s own route to the same place is the L-average Lipschitz condition of Li & Wang (2008, §5), which is why we do not attempt to repair γS . Two protocol facts that must be printed with it. First, the probe direction must be tangentprojected: a random direction understates L by a factor 5 to 21 on the same settings (column Lrand /L), because the Hessian’s variation is carried by the tangential block. On CIFAR-10 our projection is a rank-1 deflation of the score direction out of D=3072 and does essentially nothing — Lrand /L = 0.9995–1.0001, a four-digit identity — so the CIFAR L is a random-direction reading and must not be compared against κmax ; the point-cloud harness’s projection demonstrably does act, at Lrand /L = 0.048–0.187. Second, the Kantorovich L is an operator norm√over a ball and exceeds the tangential-probe reading by the plane-curve frame-rotation constant 2/ 3 = 1.1547 (measured 1.151), so the printed ℓ is low by 1.15–1.6×; the 20-of-20 verdict survives that correction with more than 3× margin. 25

Preprint: Diffusion Inverse Step as a Strongly Convex Program

What we do not print. The closed-form Newton–Kantorovich companion u(c) = (1 + c) − √ 2 c + 2c solves its own quadratic to 1.1 × 10−16 but is not the certified ℓ = 12 boundary: on the Bessel-exact circle two independent bisections of the true boundary place it a factor 1.6 to 2.2 lower and disagree with each other by 17%, and the mechanism is isolated — a and b are √ exact to four digits while L, as an operator norm over a ball of radius ≈ 2b, both picks up the 2/ 3 frame factor and reaches where L has already blown up. The law survives (the boundary is a function of c alone, σ-independent to four digits over σ ∈ [0.002, 0.02]); only the constant is wrong. We therefore print the Picard boundary wP = 1/(1 + ρ⋆g ), which is exact and verified to 1.3 × 10−4 , and no Newton companion. The enabling lemma, and Newton’s order. The third derivative of Ψt along a tangent direction is the second fundamental form: ∥D2 F [P v, P v]∥/ρ⋆g against κmax reads 0.99498/0.95814/0.95849/0.94883 (sphere / torus / trefoil / two spheres) on the exact score at σ = 0.0585 and 0.99182/0.93022/0.92491/0.91686 at σ = 0.0744; every deviation is negative, grows with σ, and lies within a factor 2.7 of (σκ)2 . This is a converging finite-σ correction, not slack, and it turns the old diagnostic into a calibrated curvature readout. Do not calibrate it on the swiss roll: it is the only class whose reading exceeds κmax (1.516 and 1.175 against the σ-law harness’s κmax = 3.2195; Table 4 normalises the same two settings by its own κmax = 3.1497 and reads 1.5591 and 1.2025), with a σ-exponent of −1.06 against −0.013 to −0.149 elsewhere, because its curvature varies threefold over the cloud so κmax is not the right supremum there. On the trained scores the same quantity is 0.021–0.124 and exactly 0.00000 in 10 of 10 rows on a ReLU backbone — the piecewise-constant-Jacobian mechanism, and the precondition Manton (2015) identifies for quadratic convergence, made visible. Consistently, Newton’s method on the inversion residual converges at order 0.964–1.464 on the CIFAR-10 UNet, not 2. We do not attribute that: our step solves the symmetrised system and discards a skew part we measure at 0.115–0.433, the linear solve is 20 CG iterations and an inexact Newton step caps the observed order at 1, and our own estimator returns only 1.631–1.849 on exactly smooth references. The prescribed SiLU-versus-ReLU activation control was not run. Table 4: The Kantorovich triple against its Bayes references. Columns a/eht , b/b⋆ and L/(ρ⋆g κmax ) each have Bayes value 1; ℓ = abL certifies at ≤ 12 . Lrand /L is the same constant read along an unprojected direction. M

score

t

a/eht

b/b⋆

L

L/(ρ⋆g κmax )

Lrand /L

ℓ = abL

sphere sphere torus torus swiss roll swiss roll trefoil trefoil two spheres two spheres

exact exact exact exact exact exact exact exact exact exact

2 3 2 3 2 3 2 3 2 3

0.9987 0.9985 0.9895 0.9875 0.9979 0.9982 0.9992 0.9993 0.9901 0.9884

0.9987 0.9985 0.9898 0.9878 0.8528 0.8898 0.9545 0.9627 0.9901 0.9884

0.2723 0.2132 1.0161 0.7773 1.3401 0.8117 0.6078 0.4641 0.7196 0.5461

0.9950 0.9920 0.9630 0.9381 1.5591 1.2025 0.9546 0.9280 0.9488 0.9167

0.122 0.122 0.092 0.092 0.048 0.057 0.073 0.072 0.122 0.123

0.00545 0.00464 0.03950 0.03300 0.08537 0.05419 0.01026 0.00858 0.03924 0.03229

sphere sphere torus torus swiss roll swiss roll trefoil trefoil two spheres two spheres

trained trained trained trained trained trained trained trained trained trained

2 3 2 3 2 3 2 3 2 3

0.7442 0.8064 0.7451 0.8080 0.7460 0.8097 0.7429 0.8051 0.7443 0.8075

0.7370 0.7978 0.7406 0.8034 0.7402 0.8021 0.7381 0.8004 0.7414 0.8042

0.0406 0.0416 0.0158 0.0162 0.0354 0.0387 0.0460 0.0466 0.0238 0.0238

0.1483 0.1935 0.0150 0.0196 0.0411 0.0573 0.0722 0.0931 0.0313 0.0400

0.070 0.075 0.186 0.187 0.077 0.070 0.063 0.064 0.094 0.091

0.00166 0.00191 0.00089 0.00105 0.00182 0.00242 0.00233 0.00283 0.00174 0.00211

CIFAR-10 img0 CIFAR-10 img0 CIFAR-10 img1 CIFAR-10 img1

trained trained trained trained

4 30 4 30

1.0426 1.0131 1.0559 1.0170

— — — —

0.03060 0.00346 0.03812 0.00503

— — — —

1.000 1.000 1.000 0.999

0.00364 0.00020 0.00441 0.00030

26

Preprint: Diffusion Inverse Step as a Strongly Convex Program

F

P RIOR W ORK , C LAIM BY C LAIM

The novelty carve (§2). Farghly et al. (2025) prove that log-domain smoothing acts only tangentially on an affine manifold; differentiating their normal block gives the 1/σt constant of (6) for an arbitrary empirical law, and their §4 argues before us that the expressed geometry is a property of the model. Four further antecedents bound our claims and we claim novelty for none: the Tweedie–Miyasawa identity (7) (Miyasawa, 1961; Kadkhodaie et al., 2024), Wenliang & Moran’s σt2 -normalised reading of a trained spectrum, the codimension-scaled window of Niyogi et al. (2011, Thm. 1), and the Picard fixed point of Hong et al. (2024); Hang et al. (2024), neither of whom computes its constant. What is new is the potential and its modulus, the schedule constant ρ⋆g , the convergence-domain law with its finite-noise correction, the ceiling assembled from Ct ⪰ 0, the sampled-point value D − δt /2, the δt identity with its exact remainder, the Hamilton–Jacobi constant, and Assumption 2 made checkable — set out claim by claim in App. F. Diepeveen et al. (2025) build a Riemannian structure from (∇2 log p)⊤ (∇2 log p), by (3) exactly σt−2 J ⊤ J, as an a-priori guarantee for a flow they train; ours is a post-hoc calibration for an arbitrary pretrained score, and the two are complementary rather than rival — the Fisher metric is affine in J (Raskutti & Mukherjee, 2015), so their normal-bundle object and the tangent-bundle metric have traces summing to D. Azeglio & Di Bernardo (2025) select an operating point by an angle criterion, a validity check of the kind we formalise. (7) and the stiffness certificate. The matrix identity is the second-order Tweedie–Miyasawa relation (Miyasawa, 1961), stated in this literature by Kadkhodaie et al. (2024, Eq. (9)) as ∇f ⋆ = I + σ 2 ∇2 log pσ = σ −2 Cov(x | y), who read its spectrum as adaptive shrinkage and establish Ct ⪰ 0. Wenliang & Moran (2022) read the SVD of a trained U-Net’s score Jacobian at several noise levels against the scale σt2 , with a local-dimension rule at a fixed tolerance. Ours is the singular-value consequence: s1 (J ⋆ ) = 1/σt exactly iff the extremal direction carries zero posterior variance, and s1 (J ⋆ ) > 1/σt as soon as some ci > 2σt2 — which neither Assumption 1 nor Ct ⪰ 0 excludes — hence Assumption 2 as a condition to be checked rather than assumed, and (8) as the closed form of the tolerance. The contraction constant. Hong et al. (2024, Eq. (7)) state a schedule-only nonexpansiveness threshold for the same Picard operator, requiring the data-prediction model to be (σti−1 αti (e−hi − 1)/σti )−1 -Lipschitz: exactly ρg ≤ 1 with the Lipschitz constant left unknown. Hang et al. (2024) prove qualitative√contraction of the identical map and conclude existence and uniqueness by Banach, with factor 1−k ᾱt for an√ unspecified k ∈ (0, 1); rearranging shows that assumption is equivalent to positing Lip(εϕ ) = (1 − k ᾱt )/σt , i.e. to positing what (7) derives. Neither computes the constant. We supply it, and ρg = ρ⋆g St separates schedule from model. Monitoring the input-Jacobian spectral radius of an implicit layer is the standard deep-equilibrium instrument (Bai et al., 2021; Pokle et al., 2022); only its reference value is new. The dimension functional. Kamkari et al. (2024) define the estimator and evaluate it at the scaled clean point; Leung et al. (2025) prove its d − D limit there for any smooth submanifold, dropping the affine assumption of earlier justifications. Yeats et al. (2025, Eq. (7), Thm. 3.3) bound the noised-sample expectation below by d. Proposition 6 evaluates that expectation exactly, for every model and every data law, and thereby explains the design rather than criticising it. The rank ceiling and the window. Stanczuk et al. (2024) state the budget requirement informally and prescribe K = 4D; p Prop. 9 is its exact √ form, √ and the centring is ours, not theirs. Niyogi et al. (2011, Thm. 1) give 8(D − d) σ < c( 9 − 8) reach(M)/9 under exactly this noise model; Farghly et al. (2025) and Yaguchi et al. (2025) carry related conditions. Only the estimator-validity reading and the product factorisation are ours, and we claim nothing for the latter. The exponent. That a Morse–Bott critical set forces the optimal Łojasiewicz exponent is proved abstractly by Feehan (2017), and the equivalence of the Polyak–Łojasiewicz, quadratic-growth and error-bound conditions with the Morse–Bott property by Rebjock & Boumal (2025), who also record that no non-constant C 1,1 function admits an exponent below 12 . Equation (13) supplies what those √ do not: the constant cL = 2/σt , from the eiconal property inside the reach. The mean-curvature 27

Preprint: Diffusion Inverse Step as a Strongly Convex Program

residual of Prop. 7 is obtained independently, in the density-estimation setting, by Chen et al. (2026, Thm. 3.4). The potential, and the constant nobody computed. Hong et al. (2024, Eq. (7)) state a scheduleonly nonexpansiveness threshold for the same Picard operator, which is exactly ρg ≤ 1 with the Lipschitz constant √ left free; Hang et al. (2024) prove qualitative contraction of the identical √ map with a factor 1 − k ᾱt for an unspecified k, which rearranges to positing Lip(εϕ ) = (1 − k ᾱt )/σt — that is, to positing what (7) derives. Neither writes the potential, and neither computes the constant. What we add is Ψt , the modulus e−ht and the factorisation ρg = ρ⋆g St+1 . Ostrowski (1973) is the standard caveat that a spectral radius below 1 does not by itself give a contraction in a fixed norm; it does not rescue undamped Picard here, because the run measures ρ(Bt J) = ρg to fifteen digits at the probe, so the two coincide and the failure is the step size. The convergence domain. The obvious referee suggestion — run the Riemannian α-theory instead — contributes nothing in this setting: Dedieu et al. (2003) give Kζ = 1 and rζ = ∞ in flat space, so the ambient curvature terms vanish identically and the manifold’s curvature can only enter through the Hessian-Lipschitz constant L, which is what the ray-Lipschitz constant LH of §4.4 computes. Manton (2015, Thm. 1, Cor. 2) supply the smoothness precondition — quadratic convergence needs a hypothesis strictly between C 2 and C 3 — which is why a ReLU backbone voids the certificate, and the run measures exactly that at 0.00000 in 10 of 10 rows (App. E). Eckart & Young (1936) is what unifies the pieces: smin (I − Bt J ⋆ ) = e−ht is a distance to ill-posedness, and it is simultaneously the Kantorovich a, the Neumann bound (an equality at Bayes), the modulus of Theorem 1 and the fold threshold of Theorem 2. Multiplicity, the fold and the 2-cycle. Smale (1985) supplies both halves of what we need: his z 3 /2 − z + 1 carries a superattracting 2-cycle and works over the reals in the same way, and his Ch. III §4 classification says a generally convergent purely iterative algorithm has no periodic sink of least period two — which is exactly the status of undamped Picard, exhibited here at the Bayes limit with Λt = 2.011–7.067 and a 2-cycle at ±0.374819 (App. B). McMullen (1987) does not apply, and we say so rather than citing it loosely: it concerns d ≥ 4 roots and rational dependence on jets, whereas inversion has one solution and contracts at infinity. The fold/flip taxonomy of Theorem 2 is standard (Kuznetsov, 2004), and the dynamical-systems literature on basins (Curry et al., 1983; Daza et al., 2016; Aguirre et al., 2009; Grebogi et al., 1983) is cited only to record what we do not do: we run no fractal basin analysis and report no basin statistic, and neither the polynomial-like renormalisation of Douady & Hubbard (1985) nor the global root finding of Hubbard et al. (2001) transfers here, since F carries no holomorphic structure and infinity is interior to the basin rather than on its boundary. Why no change of metric repairs Remark 10. The natural suggestion — learn a pullback metric that makes Et geodesically convex — is obstructed for a closed M with dim M ≥ 1. On a Hadamard-type manifold the critical set of a geodesically convex function is connected and the argmin is a retract (Udrişte, 1994, Ch. 3 Thm. 3.4, Ch. 5 Lem. 3.5), so Crit(Et ) ∼ = M would have to be a contractible deformation retract of RD , which a closed positive-dimensional M is not. Scope it honestly: the obstruction is topological, so it fails for a contractible patch and we do not claim it for the swiss roll. Existence without any hypothesis, and why we run no multistart. The radial-index certificate is Watson (1989, Lem. 3): if the outward radial component of F is non-negative on the boundary of a ball, a solution exists inside it, with no Lipschitz, contraction, manifold, reach or unimodality hypothesis — which is why it fires in 24 of 24 CIFAR and 27 of 27 CelebA-HQ settings where every other instrument needs an assumption we cannot check. Its probability-one homotopy companion is Chow et al. (1978). We deliberately do not corroborate uniqueness with a multistart experiment: index-(−1) roots have measure-zero basins for first-order methods (Lee et al., 2019), so a null multistart is not evidence of uniqueness — and Theorem 2 shows the roots born at the fold are repelling, hence Picard-invisible by construction. The Fisher metric. Diepeveen et al. (2025) build a Riemannian structure from (∇2 log p)⊤ (∇2 log p); the Fisher metric of the same family is affine in the Jacobian, 28

Preprint: Diffusion Inverse Step as a Strongly Convex Program

σt2 gF = I − σt J (Raskutti & Mukherjee, 2015). The two objects are therefore complementary rather than competing: theirs is a rank-(D − d) normal-bundle object and the Fisher metric is a rank-d tangent-bundle one, and their traces sum to D. That is why the comparison in §2 is a statement about which bundle a construction reads, not a claim that one is better.

G

D EFERRED E XPERIMENTAL D ETAIL

The passages below are the full versions of results §5 states in brief. The oracle control (§5.2). Replacing εϕ by the exact score under the identical harness, probe cloud, ˆ ◦ ) = 255.99999, 255.99999, 256.99, 246.32, 128.16 flattening and budget at σt = 0.0426 gives d(x on the sphere, two spheres, torus, swiss roll and trefoil against flattened targets 256, 256, 256, 256, 128 — the two closed, uniformly sampled classes exact to 4·10−8 relative, with seed-to-seed variation exactly zero on 30 of 30 settings. At the same budget the rank statistic recovers the true codimension on every closed class (gap ratios 12.3, 3.4, 5.7, 4.3) and the exponent split of (14) returns codimension and flattened dimension exactly on four of five. The epoch-300 model reads 382–383 on all five classes and the epoch-50 one 391–410, i.e. the ambient D. The failure to separate d=1 from d=2 is a property of the trained score, not of the readout; both oracle misses are the swiss roll’s free boundary, and Table 5 labels them so. The scale-free negative (§5.2). The decisive test needs no ground truth: any hypothesis of the form “the readout tracks nd” predicts that the d=1 manifold reads exactly half of the d=2 manifolds. Over the 12 pairwise settings that satisfy every validity guard — the trefoil against each two-dimensional class separately, never against a mean — the epoch-300 network’s ratio lies in [0.996, 1.008] with mean 1.0003, against an exact score that reads [0.496, 0.536] on the very same settings. The residual spread is not scatter but a monotone drift in σt (0.9989 → 1.0142 from t=2 to t=12), and letting each class in turn play the d=1 role, the exact score puts the true one 16 null standard deviations below the others while the trained model puts it 0.1 above. Being a ratio this is invariant to the ground truth and to any common multiplicative bias, and its window is measured rather than assumed. The flattened amplitude readouts carry no class information about d in this setting, and the scope matters: Diepeveen et al. (2025, §6.2) separate d=1 from d=2 exactly on curves and surfaces in R3 , but their sample is one point in R3 , their readout is the latent-variance spectrum of an isometry-regularised flow they train, and they evaluate at the noiseless point with no sampling estimator. Their success is what our window predicts, and it is why a scale-free exponent is the right invariant absent isometry regularisation. The trained-shell mechanism (§5.2). On every trained score the same sweep returns no shell, and that is a measurement rather than an absence. Of our 88 trained shell readouts, 21 sit at a setting where the exact score does resolve one; all 21 are null, and the other 67 carry no information because the exact score is null there too. In 83 of the 88 the two-branch estimator never resolved a second multiplier at all, and the mechanism is measured three ways: the trained Jacobian is 11.4–13.7× too weak in amplitude at the smallest setting; the multiplier field is flat, at most 0.0178 at the smallest noise and 0.7113 over all 1491 admitted probes against a divergence threshold of 1; and the enabling lemma is absent, reading 0.021–0.124 against the exact score’s 0.92–0.995 when normalised by ρ⋆g κmax , not moving with noise, and vanishing identically on a ReLU backbone whose Jacobian is piecewise constant. A score whose second derivative is a tenth of the curvature has a multiplier field that does not bend along the ray, so there is no shell to find — a negative that is scale-free, needs no ground-truth dimension, and has a closed-form Bayes value on both branches. Three qualifications are underpowered rather than negative, and App. H states each: three settings bracket a sign change across an abstention window, and the one resolvable flip is a collapse of the positive branch rather than the Bayes mechanism; at the two smallest noise levels the divergence shell is refuted while the oscillation shell is strictly unresolved; and every σt -exponent quoted is a two-point slope on one network. The descent estimator on analytic energies (§5.2). The estimator used in practice initialises at N (0, ID ), runs fixed-step gradient descent on Et and regresses log ∥∇Et ∥ on log(Et − mink Et ). By Remark 11 it must return 12 wherever the transverse spectrum is degenerate, whatever the model, 29

Preprint: Diffusion Inverse Step as a Strongly Convex Program

and Fig. 3 confirms this on analytic energies with no diffusion model at all: θ̂ = 0.5000 with R2 = 1.000000 on the Morse–Bott energy under a divergent step (ηλmax ≈ 166, the value the published hyper-parameters imply), 0.6085 on the same energy under a stable step, and 0.077 on a genuinely quartic energy whose true exponent is 34 . It is a landscape-quadraticity detector with a mis-calibrated scale, not an estimator of θ. The field estimator on the same four energies returns 0.500000, 0.500000, 0.500000 and 0.750000 — the true exponent in every case. The contraction measurement (§5.2). Where St is readable it costs one power iteration and is, by Thm. 4, the entire explanation of the measured contraction rate. At the smallest guard-clean cloud setting (σt =0.0426) the exact score reads St /S⋆t within 8·10−8 of 1 on the two closed, uniformly sampled classes, while the epoch-300 network reads 0.050–0.055 and the epoch-50 one 0.024–0.034: an 18 to 42× deficit against a reference computed in the same harness at the same setting. At the CIFAR probe ρ⋆g = 0.1180 and the Picard rate, fitted by a matrix pencil with forward passes only, is 0.1409–0.1498 over four inputs — 19% to 27% high, one-sided, exactly what ρg = ρ⋆g σt λ1 (J) with σt λ1 > 1 predicts. At stride 20 the multiplicity margin falls below 1 in 8 of 20 settings and Picard measurably diverges (ρemp = 1.0008–1.0078 in the three images the estimator does not flag) where the Bayes map contracts at 0.7134. None of this is evidence for the Bayes law: on an affine M the normal block of the noised empirical log-density is data-independent (Farghly et al., 2025, Eq. (10)), so a perfectly memorising score also reads St = S⋆t , and the certificate is one-sided. At 2562 the same nine settings return St = 4.69–9.95 in fp32 with the closed-form checks passing, quoted as a scale and not a stiffness; it locates the reported failure of inversion at that resolution in the score, not in DDIM. The field exponent on CelebA-HQ (§5.2). The image runs supply the counterexample. On pretrained CIFAR-10 and CelebA-HQ-256 — converged by any external standard — the descent protocol returns values indistinguishable from its own undertrained pole, and they are not merely inaccurate but inadmissible: a non-constant C 1,1 function admits no Łojasiewicz inequality with θ < 12 (Rebjock & Boumal, 2025, Rem. 2.21). The mechanism is the min-subtraction, which reproduces them with no model at all, returning 0.0036 (R2 = 0.81) on a perfectly quadratic energy, so we do not report θ̂ as a training certificate. What (13) supplies instead is the field-based readout calibrated in §5.1: sample Et and ∇Et along a normal ray at x◦ , taking the slope as a test of transverse quadratic structure and the intercept as σt2 λ̂ → 1. On exact scores it returns θ = 0.50000 and σt2 λ̂ = 0.9999; on pretrained CelebA-HQ-256, θ = 0.468–0.523 at t ∈ {30, 100} with R2 ≥ 0.9998 on two of three images, against a descent estimate below 0.01 on the same model, and along a random ray it does not (0.69–0.95). The third image reads 0.34 and the intercept fails everywhere at σt2 λ̂ = 2.0 to 41.4, so with N = 3 we report the slope and not the amplitude; at t = 4 the fit is near-perfect (R2 > 0.99999) at an inadmissible exponent of −0.03 to −0.24, a falsification with a clean fit rather than noise. The exact-score shells and the two cliffs (§5.2). On the exact score both shells sit where Thm. 5 puts them. At σt+1 = 0.0585 the oscillation shell reads wosc = 0.499946, 0.499997 and 0.497288 on the sphere, the two spheres and the torus against the schedule-free 12 , and the divergence shell wdiv = 0.789124 against 1/(1 + ρ⋆g ) = 0.785615, the residual +0.003509 being the predicted 1 2 2 2 σ̃t+1 ∥II∥ = 0.003440 — a two-term correction the torus confirms to 0.8%. The three classes 1 reading 2 are exactly the three whose largest principal curvature is constant over the cloud; the other two, whose curvature spreads by 2.98× and 3.39×, return a divergence cliff at wdiv /wosc = 1.047 and 1.049 where a Bayes pair must be separated by 1.571, so a single κmax is not the supremum the law asks for and they fall outside its scope rather than contradict it. The trefoil was excluded a priori: reach(M)κmax = 0.403 < 12 while ρ⋆g < 1 always, so no schedule brings its curvature shell inside its own reach. Validity windows (§5.2). Every readout of §4 has a two-sided window in σt whose upper edge depends on where it is evaluated: Assumption 1 needs that point inside the tube,√which at x◦ means σt ≪√reach(M), while a sampling estimator evaluates at xt , whose distance to ᾱt M concentrates at σt D − d, so its condition is √ σt D − d ≪ reach(M), (15) 30

Preprint: Diffusion Inverse Step as a Strongly Convex Program

stricter by a factor of 11 at D − d = 128 and not new (Niyogi et al., 2011, Thm. 1); what we add is the reading of it as an estimator-validity criterion rather than a hypothesis of a recovery guarantee. Read as a metric-tube condition it excludes every sampling estimator everywhere, too pessimistic here because our clouds are products: with the Federer reach computed after normalisation, the x◦ condition gives σt / reach = 0.04–0.21 at the smallest probed noise, (15) gives 0.48–3.37 and the product criterion 0.151 to 0.834, so all five classes are inside the window there (Apps. D and G). Its effect is to remove an excuse: the trefoil reads 2.99× its flattened target while inside the corrected window, so its failure to separate d=1 from d=2 leaves the protocol column and §5.2’s oracle control places it in the model column. Why the trained convergence shell is structurally unmeasurable. One edge is not a limitation of this run but a derived property of the probe, and is itself a result: converting a shell radius into a curvature needs the Fermi window σt+k κmax ≤ 0.1, while the shell must lie where the model has training mass, r ≲ 1.4 σt+k , and since the shells sit at w ≥ 12 the two are irreconcilable — by 3.6 to 5.6×, for any manifold, schedule or training budget — jointly satisfied in 0 of our 25 convergence settings (§6; Table 3 setting by setting). This is why every deviation in §5.2 is quoted against the exact score at matched σt and never against the closed form, and why the trained null there is a statement about that score’s second derivative rather than a report that a shell was sought at its predicted radius and found missing. On images the upper edge additionally requires reach(M), which is estimable (Aamari et al., 2019; Yaguchi et al., 2025, Thm. 3.7, Alg. 1). The product window (§5.2). Read as a metric-tube condition, (15) puts every sampling estimator outside its regime everywhere — too pessimistic here, because our √clouds are products,√for which the reach is minp reach(Mp ) and the binding scalar concentrates at 2 log n rather than D − d. With the Federer reach computed after normalisation, the x◦ condition gives σt / reach = 0.04–0.21 at the smallest probed noise, (15) gives 0.48–3.37, and the product criterion gives 0.151 to 0.834: all five classes are inside the window there. App. D audits the five reaches, with their causes, and puts the three criteria side by side. We claim nothing for the product identity, which is necessary rather than sufficient; its effect is to remove an excuse, since the trefoil reads 2.99× its flattened target while inside the corrected window, so its failure to separate d=1 from d=2 leaves the protocol column and §5.2’s oracle control places it in the model column. Prop. 6 is an independent reason to evaluate at x◦ ; the lower edge is model-side, below the scale at which St approaches S⋆t ; and a further edge binds any curvature readout taken from the residual at x◦ .

H

FALSIFIABLE P REDICTIONS AND FAILURE ATTRIBUTION

Attribution. Separating bad assumptions from bad experiments is only possible once the Bayes values are known, and only once an exact score has been run through the identical harness. Of the failures we measure: the flattened readouts’ inability to separate d=1 from d=2 is structural, survives every change of target, and the oracle control at the prescribed budget now proves it, because the exact score reads 0.505 on the same ratio the trained score never brings below 0.996; the rank saturations are budget artefacts with a known repair, and raising K moves artefacts in both directions rather than only in ours; the out-of-window readings are protocol errors with a computed fix; and the image failures remain unattributable, but for a measured reason rather than an unknown one — along the score ray the posterior covariance never exceeds 1.12σt2 , so there is no shell whose σ-exponent could √ be taken, and the live question is now the base point rather than the reach, since x◦ sits D standard deviations inside the training noise shell. Table 5 gives one row per measured failure. Every verdict of model is licensed by an oracle reading in the same harness at the same setting; untested means the instrument never met an input it could decide. The predictions, restated with their verdicts. H1: the gap |St − Sexact | closes with training t at every noise level, and closes last at the smallest σt . Confirmed end to end — the gap closes from first to last snapshot in 51 of 55 (class, t) settings, the four exceptions all at t=64 where the model already overshoots the exact score at epoch 5 — and refuted as a monotonicity: it closes monotonically in 3 of 55, and St is monotone increasing at fixed t in 5 of 55, the sphere at t=2 reading 0.0500, 0.0447, 0.0333, 0.0240, 0.0215, 0.0165, 0.0548 at epochs 5–300. The front σ ⋆ (epoch) — the smallest σ at which the model reaches half the exact stiffness — moves inward 31

Preprint: Diffusion Inverse Step as a Strongly Convex Program

Table 5: Attribution. Verdicts are estimator, model, budget, window or untested. “Exact” is the analytic or oracle score under the identical harness, probe cloud, flattening and budget. The three hypothesis-free rows assume no manifold, reach or unimodality, so the windows of §5.2 do not apply to them and none may be discounted as out of regime. Image entries are certified lower bounds. A dash is “not run”, never “returned nothing”. readout

Bayes value

exact score

trained

verdict

ˆ ◦ ), flattened d(x

256 / 128

382–383

model

trefoil ÷ 2-D ratio rank gap, K = 4D rank gap, K = 1.08D ν split ν, swiss roll

0.500 index 128 index 128

255.99999 / 128.16 0.5049 12.31 at 128 not run

0.9989 1.16 at 373, abstains 5.06 at 128

model model budget

128 / 256 128 / 256

128 / 256 (4 of 5) 174 / 210

n+ = 0–46 —

St /S⋆t

1

0.050–0.055

wosc , wdiv

1 , 1/(1+ρ⋆g ) 2

none in 21 of 21

model

wosc , swiss / trefoil

1 2

0.9999999– 1.0000000 0.499946, 0.789124 cliff at 0.19 / 0.14

model window (free boundary) model (18–20×)

—

∥D2 F [P v, P v]∥/ ρ⋆g κmax σt λmax (sym J)

1

0.92–0.99

0.021–0.124; ReLU 0.000

window (κ spread 3×) model

≤1

1.0000003

model, hyp.-free

skew index Q

0

≤ 2 × 10−6

σt tr J/D branch attribution

≤1 curvature vs bottleneck 2σt2 at the shell

— 2 correct, 1 abstain

1.011–1.132; images 1.26–4.66 0.003–0.045; images 0.11–0.43 1.12–1.74 on 12 of 16 no shell to attribute ≤ 1.12σt2 on the score ray

untested (base point)

image shells, λmax (Ct )

no oracle at D=3072

model, hyp.-free model, hyp.-free untested

by 2.15/2.29/2.16/2.21/2.09× on the five classes, but non-increasingly at every step on only three. The correct form of H1 is a front, not a per-setting monotonicity, and we restate it as such. H2: on the exact score every entry of Table 1 reproduces. Confirmed, with two closed-form corrections that are themselves the content: the S⋆ reference must be the per-point minimum of the shape-operator invariant rather than its mean over the manifold, which moves the swiss roll from the shipped S/S⋆ = 1.0019 to 0.99999924 and the torus from +7.1 × 10−4 to −1.4 × 10−4 — both an offline recomputation substituting the per-point Qmin for the mean used in the original grid, not a reported column — and the residual deviations of the two classes with non-constant curvature are exactly the amount their reference was mis-specified. H3: ρemp grows with D only through St . Not run; it needs a matched-St pair across two ambient dimensions, which this run does not contain. H4: a patch-wise analogue of (14) reads far below the patch dimension. Half-confirmed, with a 36% dead band: on an 8 × 8 patch at (12, 12) in the full 3P 2 = 192 basis, ν > 12 in 124 of 192, so dˆ = 68, which is 35.4% of the patch dimension rather than “≪”; within ±0.25 of the two Bayes values there are 108 at ν=1, 15 at ν=0 and 69 unresolved. Under a resolved-only reading the tangential count is 7.8% and H4 holds; under the midpoint rule it does not, and one patch on one image at one noise pair cannot separate the two readings. The 64 stiffest directions also cluster at 0.886–0.916 (mean 0.902) against a Bayes normal exponent of exactly +1 — a systematic 8–11% shortfall with a spread of only ±1.5%, unexplained. H5: rank readouts track their budget. Confirmed as stated and refuted as a hope. At K = 4D no trained setting reproduces the codimension-128 reading we obtained at K = 1.08D (sphere converged rmax = 1.157 at index 373; sphere undertrained 1.028 at index 383), so raising K retired our own positive; the positive moves to the exact score, at 12.307 at index 128. Symmetrically, the unthresholded difference rule emits a confident dˆ ≈ D − 2 on trained models and the ratio rule returns a confident wrong answer on the one class with a boundary (App. D). Both rules fail on trained scores; only the corner differs, and we claim neither. 32

Preprint: Diffusion Inverse Step as a Strongly Convex Program

W HAT THE RUN DID NOT SETTLE Stated compactly in §6: Three things we predicted and could not measure, each stated in App. H in the form we are willing to defend. Whether the convergence-domain law is observable on any trained score is open, and our own measurement of why it is not observable here — a Hessian–Lipschitz constant at 2–12% of the curvature the law reads — is a two-point σ-fit on one network with no error bar. The multiplicity certificate is a theorem with a threshold verified in an analytic model and zero observations on a trained score anywhere in this study. And the image failures stay unattributable: 2 along the score ray the posterior covariance never exceeds √ 1.12σt , so there is no shell whose σexponent could be taken, and the point Prop. 6 forces sits D standard deviations inside the training noise shell — the live confound, in place of the unknown reach. Our two training states are snapshots of one run and our two networks two draws of one recipe, differing in St by a median 3.0% and a maximum 13.8%, so no trained class-to-class comparison here carries a population error bar, only an estimator resolution and a network-to-network spread, and we name which we mean wherever we quote one. Finally, we claim the focal-radius law for the Bayes limit and the oracle only: the multiplier statement holds at every σt and for every data law, but the finite-noise correction and any radius-to-curvature conversion hold only inside the Fermi window, and the law is observable only where κmax is roughly constant over the probed set — the two classes whose curvature spreads threefold return a divergence cliff at wdiv /wosc = 1.05 against the 1.57 a genuine Bayes pair must show, a hypothesis of the measurement rather than a failure of the theorem.

33

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