Hessian Geometry of Latent Space in Generative Models

Alexander Lobashev; Dmitry Guskov; Maria Larchenko; Mikhail Tamm

REVIEW 4 major objections 5 minor 2 cited by

Hessian Geometry of Latent Space in Generative Models

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A Fisher metric recovered by learning the log-partition function reveals fractal phase boundaries in diffusion-model latent spaces, where geodesic interpolation breaks down.

desk verdict The Ising/TASEP reconstruction is a genuinely new and credible piece of work, but the diffusion-model phase transitions are an artifact-prone leap that the theory does not cover. read the letter →

arxiv 2506.10632 v1 pith:Y2WWKYAF submitted 2025-06-12 cs.LG cond-mat.stat-mechcs.CVmath.DGmath.STstat.TH

Alexander Lobashev , Dmitry Guskov , Maria Larchenko , Mikhail Tamm This is my paper

classification cs.LGcond-mat.stat-mechcs.CVmath.DGmath.STstat.TH MSC 62B1153B1268T07

keywords FisherinformationmetricHessiangeometrylatentspacediffusionmodelsphasetransitionsexponentialfamilylog-partitionfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The authors aim to establish that the Fisher information metric of a generative model's latent space can be reconstructed from generated samples alone, by training a network to learn the log-partition function of the posterior over latent parameters. If their reconstruction is correct, the latent space of a text-to-image diffusion model is not smooth: it splits into distinct phases separated by fractal, self-similar boundaries where the recovered metric loses smoothness and the generative map's Lipschitz constant diverges. Inside a single phase, geodesic interpolation is nearly linear; crossing a boundary, it becomes nonlinear. The method is first validated on exactly solvable Ising and TASEP models, where the reconstructed free energy matches the exact solution and beats two baselines, then applied to two-dimensional slices of a diffusion model.

What carries the argument

The log-partition function $\log Z(t)$ (the log of the normalizing constant of $p(x|t)$, viewed as a function of the latent parameter) is the central object: for exponential families its Hessian $\nabla^2 \log Z(t)$ equals the Fisher information metric. The paper's two theorems make it learnable from samples: Theorem 3.1 identifies the $N$-sample posterior limit with an exponential of the negative Bregman divergence (the gap between $\log Z(t)$ and its first-order Taylor approximation at $t'$), and Theorem 3.2 shows that matching that exponential determines the Hessian. The Bryant-Amari-Armstrong theorem supplies the bridge to non-exponential generative models on two-dimensional latent slices.

What would settle it

Estimate the actual distribution distance between generated images at nearby latent points (for example with a learned likelihood model or by comparing many sample pairs) and compare it with the squared feature-embedding distance used in the paper; if the two disagree near the alleged phase boundaries, the reported boundaries are artifacts of the feature encoder rather than geometry of the generative model.

Watch

Extended reading notes

Core claim

For an exponential family $p(x|t) = e^{\langle t, f(x)\rangle - \log Z(t)}$, the paper proves that the posterior over parameters given $N$ samples from $p(x|t')$ concentrates as $\bigl(p(t|x_1,\dots,x_N)\bigr)^{1/N} \to e^{-D_{\log Z}(t,t')}$, where $D_{\log Z}$ is the Bregman divergence, and that matching this exponential against the approximated posterior recovers $\log Z$ up to an affine term. Since the Fisher metric of an exponential family equals the Hessian $\nabla^2 \log Z(t)$, this gives a sample-based route to the latent-space metric. The Bryant-Amari-Armstrong theorem, stating that any two-dimensional analytic Riemannian metric locally admits a Hessian representation, extends the route to two-dimensional slices of non-exponential generative models. Applied to a text-to-image diffusion model with a pre-trained contrastive image encoder used as a stand-in for KL divergence, the recovered $\log Z$ shows abrupt derivative changes and fractal phase boundaries; geodesics are nearly linear within a phase and fail at boundaries, and Proposition 4.1 exhibits a bimodal Gaussian mixture whose reverse-ODE Lyapunov exponent $\lambda = \frac{\beta}{2}\left(1 + \frac{1-\sigma^2}{\sigma^4}\right)$ diverges as $\sigma \to 0$, modelling a boundary point.

Load-bearing premise

The diffusion-model results assume that the distance between feature embeddings of two generated images accurately measures how different the underlying image distributions are, with the feature distribution treated as Gaussian with identity covariance; if that proxy fails, the reconstructed metric and its phase boundaries describe the feature extractor rather than the generative model.

Editorial extensions

If this is right

Latent-space interpolation should be treated as piecewise-geodesic: ordinary linear interpolation is adequate within a phase, but meaningful interpolation across a phase boundary requires the Fisher-metric geodesic.
The same reconstruction pipeline can be applied to any two-parameter family of generative models, providing a way to detect phase transitions and locate non-smooth regions without knowing the model's unnormalized density.
The fractal, self-similar structure of diffusion phase boundaries implies that arbitrarily small latent perturbations near a boundary can switch the generated image between distinct visual modes, down to the numerical precision of the latent representation.
Adding stochasticity to the diffusion sampler smooths the reconstructed free-energy landscape and suppresses sharp phase boundaries, so the phase structure is most cleanly observed in the deterministic sampling regime.
Free energy can be reconstructed without the Hamiltonian, which the paper shows by matching exact solutions on Ising and TASEP and beating the posterior-mean-as-statistics and PCA-VAE baselines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The mechanism implied by Proposition 4.1 suggests a general picture: the reverse diffusion map stretches a unimodal latent distribution onto disjoint image manifolds, so phase boundaries are exactly the places where nearby latents land on different manifolds; this picture could be tested by checking whether boundary locations predict the onset of multi-modality in generated sample sets.
Because the diffusion results depend on the choice of image encoder, a natural extension is to validate the reconstructed metric with multiple feature extractors and see whether the phase boundaries are invariant; if they are not, the boundaries are properties of the embedding rather than of the generative model.
The divergent Lipschitz constant at boundaries could be exploited practically: it offers a principled way to detect which latent directions are semantically unstable and to build interpolation or editing algorithms that route around phase boundaries.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. The core idea—learn the log-partition from an approximate posterior and read off the metric as its Hessian—is genuinely new, and the Ising/TASEP experiments are the strongest part of the paper. But the diffusion-model phase boundaries, which are the headline, are not supported: the metric is the Hessian of a function fit to CLIP embedding distances, not to the generative model's likelihood, and the convergence theorem is proved only for the L2 loss the authors admit they don't use in practice.

What is actually new: the Bregman/log-partition framing gives a principled way to get a Fisher metric from samples alone, and applying that to diffusion latent spaces to look for phase structure is new. The Ising and TASEP experiments are careful, use exact or high-quality reference solutions, and beat the posterior-mean and PCA-VAE baselines by a wide margin on derivative reconstruction. That part is worth reading.

The problems start when the method leaves exponential families. Theorems 3.1 and 3.2 are proved for exponential families with an L2 loss; the actual experiments use a JSD loss with no convergence guarantee. For TASEP, the distribution is not exponential, and the appeal to Bryant–Amari–Armstrong only tells you a Hessian representation exists—it doesn't show the potential fit to the posterior is the log-partition of the model. For diffusion, the posterior is replaced by exp(-N/2 ||CLIP(x)-CLIP(x')||^2). That would be valid only if CLIP is a sufficient statistic and feature distributions are Gaussian with identity covariance; neither is tested. The U2-Net experiment gives a smooth log Z while CLIP gives fractal cusps—pretty direct evidence that the phase structure is feature-extractor dependent, not intrinsic to the generative map. The paper even claims phase boundaries were stable across feature extractors while showing the opposite in the same section. The toy bimodal-Gaussian proposition is about data-space x=0, not the latent-space coordinates used in the experiments, so it doesn't carry the weight assigned to it.

Where this leaves us: the solvable-model part is a credible contribution, and the diffusion section is a speculative application built on unverified assumptions. The abstract's claim of fractal phase transitions and divergent Lipschitz constant in the latent space of diffusion models is not established. The paper deserves a serious referee because the method is interesting and the code is available, but it needs major revision: either validate the CLIP proxy as an approximation to the model's likelihood, or reframe the diffusion results as an analysis of CLIP-induced geometry rather than Fisher geometry of the generative model. My recommendation: send it to review with a referee who knows both information geometry and diffusion models, and expect a substantial rewrite of the diffusion half.

Referee Report

4 major / 5 minor

Summary. The paper proposes a two-step procedure for reconstructing the Fisher information metric on the latent or parameter space of a generative model: first approximate the posterior p(t|x1,...,xN) over parameters given generated samples, then train an MLP to represent the log-partition function log Z(t) by matching this posterior, with the metric given by the Hessian ∇²log Z(t). Theoretical guarantees are stated for exponential families (Theorems 3.1 and 3.2). The method is validated against exact free energies for the 2D Ising model and TASEP, and then applied to two-dimensional slices of a Stable Diffusion latent space using CLIP embedding distances as a proxy for KL divergence. The diffusion experiments report fractal phase boundaries, approximately linear geodesics within phases, and a divergent Lipschitz constant at boundaries (Sec. 4.2, Proposition 4.1).

Significance. The idea of reconstructing a Hessian metric from posterior estimates is attractive, and for exponential-family models with a strictly convex log-partition function the posterior concentration argument of Theorem 3.1 is essentially correct. The code release and the comparison against exact Ising and TASEP free energies are useful assets. However, the paper's central diffusion-model claim—that fractal phase boundaries are intrinsic Fisher-geometric properties of the latent space—is not established. The experiments replace the generative likelihood with CLIP embedding distances under sufficient-statistic and Gaussian assumptions that are not validated, and the deterministic DDIM sampler makes the Fisher metric undefined without additional stochasticity. The sign error in Lemma 3.3 and the unproven extension to non-exponential families (TASEP) further weaken the theoretical foundation, while the paper contradicts itself on whether the phase structure is stable across feature extractors.

major comments (4)

[§3.2, Eq. (21), Lemma 3.3] Lemma 3.3 and Eq. (21) contain a sign error: the normalized density derived from exp(−D_φ(t,t′)) should be exp(⟨t,∇φ(t′)⟩−φ(t)) / ∫ exp(⟨s,∇φ(t′)⟩−φ(s))ds, but the paper instead uses exp(−⟨t,∇φ(t′)⟩+φ(t)) in the numerator and denominator, which is the reciprocal of the correct kernel. Consequently, the Jensen–Shannon loss in Eq. (22) does not compare the estimated posterior with the correct normalized Bregman posterior, and the minimizer of L1(θ) need not recover log Z up to an affine transformation. Theorem 3.2, which is proved only for the L2 loss of Eq. (13), therefore does not justify the loss actually used in the experiments.
[§2.2, §4.1] The method's convergence theorem (Theorem 3.1) is proved only under the exponential-family assumption stated in Eq. (9). Section 2.2 explicitly notes that TASEP does not take the exponential-family form of Eq. (6), yet the paper presents TASEP as a validation in Sec. 4.1 and Fig. 4. The invocation of Theorem 2.1 (Bryant–Amari–Armstrong) does not fill this gap: that theorem only asserts the existence of a local Hessian representation for any analytic 2D metric, and it says nothing about the posterior-based estimator of Eqs. (19)–(22) converging to the Fisher metric. The TASEP result is therefore without theoretical support, and the sentence in Sec. 2.3 claiming the approach is theoretically justified for any generative model is not supported by the stated theorems.
[§4.2] The diffusion experiments fix DDIM η=0, making the generative mapping deterministic. For a deterministic map, p(x|t) is a point mass, so the Fisher metric of Eq. (4) is not defined without additional stochasticity; the paper itself defines the framework in Sec. 2 in terms of a stochastic generative mapping. The posterior in Eq. (19) is instead constructed from CLIP embedding distances, relying on the assumptions in Eqs. (16)–(18) that CLIP is an approximate sufficient statistic and that feature distributions are Gaussian with identity covariance. These assumptions are neither tested nor derived from the diffusion model. The learned Hessian in Eq. (24) is hence the Hessian of a surrogate CLIP-embedding process, and the paper never establishes that this equals the Fisher metric of p(x|t). The fractal boundaries and divergent Lipschitz constant reported in Sec. 4.2 and Proposition 4.1 are therefore not shown to be properties of the diffusion model's latent-space geometry.
[§4.2] Section 4.2 contains a direct contradiction on the robustness of the phase boundaries. In the paragraph 'Using a Unet Mapping', the paper states that U2-Net predicts the generation parameters exactly and yields a smooth log Z(α,β), while in the paragraph 'Baselines and Metrics' it claims that 'the phase boundaries remained stable across all tested feature extractors.' Since U2-Net is one of the tested feature extractors, these statements cannot both hold; the comparison actually demonstrates that the reported fractal structure depends on the feature representation rather than being an intrinsic property of the diffusion model. This internal inconsistency directly undermines the central claim of the paper.

minor comments (5)

[Throughout, Sec. 4.2] The name 'Lipschitz' is misspelled as 'Lipshitz' in several places, including Sec. 4.2 and the Discussion.
[Appendix A, Eq. (47)] In the proof of Theorem 3.1, Eq. (47) writes e^{-D_B(t',t)} with the Bregman divergence arguments reversed relative to the theorem statement; the final conclusion is correct, but the intermediate notation is confusing and should be corrected to match Eq. (11).
[§3.2] The statement that the MLP 'converges to the convex function during the training' is unsubstantiated; no convergence result for the nonconvex MLP optimization is provided.
[§4.2, Eq. (19)] The paper does not specify which CLIP variant, feature normalization, or dimensionality reduction is used when computing the distances in Eq. (19), making the diffusion experiments difficult to reproduce.
[Fig. 6] The caption claims self-similarity down to float16 precision at scale 10^{-8}, but no scale bar or axes description is provided to verify the claimed magnification levels.

Circularity Check

1 steps flagged · score 6.0 of 10

The diffusion-model phase transitions are, by construction, the cusps of the CLIP-distance surrogate posterior: Eq. 19 defines the target posterior from CLIP embeddings, Eq. 22 fits log Z_theta to it, and Eq. 24 declares its Hessian to be the Fisher metric, so the fractal boundaries are re-descriptions of the input feature distances.

fitted input called prediction [Sec. 3.1 Eq. 19; Sec. 3.2 Eqs. 22 and 24; applied in Sec. 4.2 (Figs. 5-6)]
"Then the posterior could be approximated as p(t|x1, . . . , xN ) ≈ e− N 2 ||E(x)−E(x′)||2 , x ∼ p(x|t), x′ ∼ p(x|t′). (19) ... The resulting approximation of the Fisher metric is gF (t) = ∇2 log Zθ∗ (t), θ∗ = argmin θ L(θ) (24)"

Equations 22 and 24 make the diffusion "Fisher metric" the Hessian of a potential that is fit, by construction, to the CLIP-distance posterior of Eq. 19. The loss L1(θ) drives p_logZ_θ(t|t') toward p(t|x1,...,xN) ≈ exp(−N/2 ||E(x)−E(x')||²), so any cusp or fractal structure already present in the CLIP embedding-distance field must be encoded in ∇² log Z_θ*; the phase boundaries of Fig. 6 are therefore a re-description of the input feature distances, not an independent property of StableDiffusion's p(x|t). The chain Eqs. 16-18 that would identify the CLIP posterior with the true posterior (E as approximate sufficient statistic, Gaussian identity-covariance features) is assumed, not demonstrated. The U2Net result in Sec.

full rationale

We examined the derivation chain from Eq. 8 through Theorems 3.1 and 3.2 to the diffusion experiments. The statistical-physics core is independent: Ising is an exponential family, TASEP has an exact free energy, and the reconstructions are compared to Onsager/exact TASEP solutions (Table 1, Figs. 3-4); no fitted input is renamed as a prediction there. The circularity is confined to the diffusion claim. In Sec. 3.1 the posterior is replaced by Eq. 19, a Gaussian-in-feature-space function of CLIP distance. The training loss Eq. 22 fits log Z_θ so that its normalized Bregman exponential matches this CLIP posterior, and Eq. 24 then identifies ∇² log Z_θ* as the Fisher metric. Therefore the recovered metric is, by construction, the Hessian of the CLIP-distance surrogate; any discontinuity or fractal cusp in the CLIP distance field must appear as a phase boundary in the learned metric. The paper's own U2Net experiment (Sec. 4.2) shows that a different posterior estimator yields a smooth log Z, confirming the feature-extractor dependence. The conditions that would make the surrogate equal the true Fisher metric of p(x|t) (Eqs. 16-18: E approximately sufficient, Gaussian identity-covariance features) are stated but never validated for StableDiffusion/CLIP. Thus the headline result for diffusion models is not an independent measurement of the generative model's Fisher metric; it is the fitted CLIP-distance potential renamed as a phase transition. No self-citation chain is involved; the Bryant-Amari-Armstrong theorem is external, though it only ensures existence of a Hessian representation and is not what supplies the experimental equality. Score 6 reflects partial circularity: the diffusion prediction reduces by construction, while the exact-solvable-model validation remains independent.

Assumptions & free parameters 5 free parameters · 6 assumptions · 2 invented entities

The central claim relies on many unvalidated assumptions for the diffusion experiments: exponential-family structure (not satisfied for TASEP or diffusion), CLIP as sufficient statistic, Gaussian identity-covariance features, and unproven convergence of the practical JSD training. The free parameters and ad hoc assumptions above show that the diffusion discovery is not a parameter-free derivation.

free parameters (5)

posterior smoothing width sigma = 1/50 (Ising), 1/150 (TASEP)
Chosen by hand in Appendix B.3 to define p_target(t|t') = C exp(-||t-t'||^2/(2 sigma^2)). It sets the effective sample size N ~ 1/sigma^2 and controls the sharpness of the recovered metric.
effective sample count N in CLIP posterior = not stated
Eq (19) uses N in exp(-N/2 ||E(x)-E(x')||^2), but the paper never specifies N for the diffusion experiments; this directly sets the posterior width.
CLIP feature covariance = identity (assumed)
Eq (18) assumes p(E(x)|t) = N(mu_i, I). The identity covariance is not estimated; it is an ad hoc choice determining the posterior and metric.
MLP architecture for log Z = 5 hidden layers, width 512, ReLU
Architecture chosen for the log-partition approximator; not load-bearing but arbitrary.
diffusion generation settings = guidance scale 5, 50 DDIM steps, eta=0, latent normalization
Hyperparameters of the diffusion sampler that define the generative mapping; phase structure may depend on them.

assumptions (6)

domain assumption p(x|t) is an exponential family for the model being analyzed
Theorem 3.1 requires this. Ising satisfies it; TASEP does not (Sec 2.2), and diffusion models do not. The Bryant-Amari-Armstrong theorem is invoked to extend to non-exponential models, but it does not guarantee the fitted Hessian equals the true Fisher metric.
standard math Bryant-Amari-Armstrong theorem: every 2D analytic Riemannian metric is locally Hessian
Used in Sec 2.3 and Discussion to justify applying the Hessian method to 2D sections of arbitrary generative models. Standard theorem, but it guarantees existence of a local Hessian potential, not that the posterior-fitting procedure recovers it.
ad hoc to paper A pre-trained feature extractor E acts as an approximately sufficient statistic: D_KL(p(x|t1)||p(x|t2)) is approximated by d(E(x1),E(x2))
Equations (16)-(17) in Sec 3.1. No verification that CLIP is sufficient for the diffusion model; the entire diffusion posterior approximation depends on this.
ad hoc to paper Feature distributions p(E(x)|t) are approximately Gaussian with identity covariance
Equation (18) in Sec 3.1. This yields the squared-distance posterior in Eq (19). Not validated; if false, the recovered Fisher metric is not the metric of the generative model.
domain assumption Uniform prior p(t) over the latent domain S
Equation (15); standard choice to avoid bias toward specific regions.
ad hoc to paper The MLP trained with JSD loss converges to the true log Z(t)
Sec 3.2 states that the MLP converges to the convex function during the training but provides no proof. The proven convergence (Theorem 3.2) is for an L2 loss, not the JSD loss in Eq (22).

invented entities (2)

Phase and phase-transition boundary in diffusion latent space
purpose: Describe abrupt changes in the reconstructed Fisher metric and image appearance during latent interpolation
The paper qualitatively shows image switching (mountain vs lion) and claims self-similar fractal structure, but provides no quantitative definition of phase, no fractal dimension, and no null-model comparison. The boundaries are features of the fitted log Z, not independently confirmed properties of the generative model.
Divergent Lipschitz constant at phase boundaries
purpose: Characterize high sensitivity of diffusion output to small latent changes near phase boundaries
Proposition 4.1 computes a divergent Lyapunov exponent for a toy bimodal Gaussian mixture, not for the actual Stable Diffusion model. The paper does not measure a Lipschitz constant for the actual model; the claim is inferred from qualitative observations.

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Cite this review

Pith. "Pith review of Hessian Geometry of Latent Space in Generative Models." pith.science (2026). https://pith.science/paper/Y2WWKYAF

@misc{pith2026250610632,
  author       = {Pith},
  title        = {Pith review of: Hessian Geometry of Latent Space in Generative Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2WWKYAF}},
  note         = {Machine review of arXiv:2506.10632}
}

read the original abstract

This paper presents a novel method for analyzing the latent space geometry of generative models, including statistical physics models and diffusion models, by reconstructing the Fisher information metric. The method approximates the posterior distribution of latent variables given generated samples and uses this to learn the log-partition function, which defines the Fisher metric for exponential families. Theoretical convergence guarantees are provided, and the method is validated on the Ising and TASEP models, outperforming existing baselines in reconstructing thermodynamic quantities. Applied to diffusion models, the method reveals a fractal structure of phase transitions in the latent space, characterized by abrupt changes in the Fisher metric. We demonstrate that while geodesic interpolations are approximately linear within individual phases, this linearity breaks down at phase boundaries, where the diffusion model exhibits a divergent Lipschitz constant with respect to the latent space. These findings provide new insights into the complex structure of diffusion model latent spaces and their connection to phenomena like phase transitions. Our source code is available at https://github.com/alobashev/hessian-geometry-of-diffusion-models.

Figures

Figures reproduced from arXiv: 2506.10632 by the authors.

**Figure 1.** Visualization of Theorem 3.1. Having samples from p(x|t ′ ) we can approximate posterior distribution p(t|x1, . . . xN ). The partition function log Z(t) defines a Hessian metric on the latent space. Our Theorem 3.1 guarantees that log Z(t) in limit depends only on the posterior distributions p(t|x1, . . . , xN ) and Theorem 3.2 offers a way to learn log Z(t) from p(t|x1, . . . , xN ). community from two different… view at source ↗

**Figure 2.** Visualizing Bregman divergence. (Top) A convex function log Z(t) (black curve) and its tangent line (blue) at a point t ′ . The Bregman divergence DB(t, t′ ) (vertical green segment) measures the difference between log Z(t) and the linear approximation at t ′ . (Bottom) The exponential distribution e −DB(t,t′ ) (blue surface) approximates the posterior distribution on parameters p(t|x1, . . . , xN ) 1 N , where x1,… view at source ↗

**Figure 3.** 2D Ising model. Partial derivative of the recon [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: TASEP. Left: reconstructed free energy (red) com [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Free energy of diffusion model reconstructed with [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: The fractal structure of phase boundary in the interpolation landscape of diffusion model. The last plot represent [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: (A) Geodesic (ours) and linear interpolation between images. Note that our interpolation variant is more [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Illustration of Proposition 4.1 operate in more general settings than MCMC-based techniques like importance sampling. Bryant-Amari-Armstrong theorem allows us to go beyond exponential family distributions and apply our method to study two-dimensional sections of laten…

**Figure 9.** Figure 9: Free energy surface of a diffusion model ( [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

**Figure 11.** Figure 11: Training examples for free energy reconstruction. Ground truth images with known generation parameters. [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

**Figure 12.** Figure 12: First- and second-order derivatives of the diffusion model’s free energy. Discontinuities mark first-order phase [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]

**Figure 13.** Figure 13: Free energy derivatives of diffusion model’s free energy reconstructed with clip distance. Overlaid grid highlights [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]

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