Repulsive Landscape Sharpening

• Repulsive landscape sharpening is a set of mechanisms that modify energy landscapes to create distinct ridges, barriers, and local minima, thereby promoting multimodal exploration.
• It spans diverse applications—from Bayesian prompt learning and blind deconvolution to tissue interface modeling and string compactifications—by employing repulsive terms to regulate system behavior.
• These methods leverage explicit energy penalties, local curvature adjustments, and consistency constraints to enhance posterior coverage, mitigate overfitting, and achieve precise outcomes in complex systems.

Searching arXiv for the cited papers to ground the article in current records. Attempting to retrieve the referenced arXiv entries for verification. Repulsive landscape sharpening denotes a family of mechanisms in which a landscape is modified so that nearby trajectories, modes, or interface configurations are actively pushed apart, thereby creating ridges, barriers, cusp-like restoring forces, or stability exclusions that suppress collapse to an undesired state and promote separation among competing solutions. In the materials considered here, this theme appears in Bayesian prompt learning, blind deconvolution with diffusion priors, interfacial fluctuation theory in dense cellular sheets, flux-vacuum selection in string compactifications, and nonequilibrium ordering in repulsive mixtures. The common thread is not a single formalism but a recurrent geometric effect: repulsion restructures the accessible landscape so that multimodal exploration, sharp local minima, or sharpened interfaces become more prominent than diffuse, collapsed, or globally attractive alternatives (Bendou et al., 21 Nov 2025, Nguyen et al., 4 Aug 2025, Yue et al., 2024, Ishiguro et al., 2021, Santos-Flórez et al., 2020).

1. General mathematical motif

A recurring formulation is the addition of a term that penalizes proximity in a space judged to be physically or functionally meaningful. In Bayesian prompt learning, the composite energy is

$$E_{\mathrm{rep}}(\Theta) = \sum_{k=1}^K E(\theta^k) + \lambda \sum_{k<l} \Phi(P_{Z^k}, P_{Z^l}),$$

where $\Phi$ is a probability divergence or distance such as Maximum Mean Discrepancy or Wasserstein distance, and the repulsive term is computed on representation distributions induced by different prompts. In that setting, repulsion explicitly separates prompt particles in representation space rather than only in parameter space (Bendou et al., 21 Nov 2025).

In blind deconvolution with diffusion priors, the operative contrast is between globally attractive and locally repulsive directions. The negative log-posterior is

$$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$

and the key empirical inequality

$q(h \star x) \le q(x)$

makes blur attractive to the prior. However, under the identifiability conditions

$$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$

the Hessian at $(\bar{x},\bar{\theta})$ is positive-definite, so the local geometry becomes sharply stabilizing around realistic sharp solutions (Nguyen et al., 4 Aug 2025).

In shape-based tissue models, sharpening is encoded by a non-analytic interfacial energy density

$\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$

The $|h'|$ term produces a cusp-like restoring force proportional to $\mathrm{sign}(h')$, which penalizes any nonzero slope with finite magnitude even for infinitesimal perturbations. This differs qualitatively from capillary-wave theory, where the restoring force is harmonic and scales linearly with amplitude (Yue et al., 2024).

This suggests that “repulsive landscape sharpening” is best understood not as generic steepening, but as a structural alteration of the effective geometry: separation terms, cusp terms, or admissibility constraints reshape the set of accessible basins and the routes by which optimization or dynamics reaches them.

2. Representation-space repulsion in Bayesian prompt learning

In Repulsive Bayesian Prompt Learning, prompt optimization is framed by the posterior

$$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$

with energy

$\Phi$0

For CLIP-style multi-modal prompt learning, the loss is the negative log of the softmax over cosine similarities,

$\Phi$1

ReBaPL maintains $\Phi$2 parallel prompt particles and uses SGHMC with a cyclical step-size schedule. In velocity form,

$\Phi$3

while the cyclical schedule alternates exploration and exploitation by keeping the first $\Phi$4 steps near $\Phi$5 and the last $\Phi$6 steps near $\Phi$7 (Bendou et al., 21 Nov 2025).

The distinctive sharpening mechanism is the representation-space repulsive potential. Each particle induces a mini-batch representation set

$\Phi$8

with empirical distribution $\Phi$9. The repulsive composite energy adds

$$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$0

and in the inter-cycle version the current cycle’s particles are repelled from the last cycle’s stored particles. The update becomes

$$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$1

The paper also uses a potential

$$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$2

with force $$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$3, and setting $$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$4 as a representation-space distance recovers the same effect (Bendou et al., 21 Nov 2025).

For $$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$5, the method uses MMD or Wasserstein distance. The MMD estimator is

$$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$6

typically with an RBF kernel. Wasserstein variants include exact empirical OT, sliced Wasserstein, Sinkhorn OT, and Gaussian $$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$7 (Bendou et al., 21 Nov 2025).

The reported effect is that sampling multiple prompt configurations while actively pushing their induced representations apart creates “ridges” and “barriers” between basins of attraction, so the sampler does not prematurely collapse to a single solution. Empirically, on base-to-novel generalization over 11 datasets in the 16-shot setting, MaPLe + ReBaPL improves average harmonic mean by $$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$8 points and MMRL + ReBaPL improves average harmonic mean by $$\ell_y(x,\theta) = \frac12 \|H_\theta x - y\|_2^2 + \sigma_y^2 q(x),$$9 points; ablations show that for MaPLe the harmonic mean improves from $q(h \star x) \le q(x)$0 to $q(h \star x) \le q(x)$1 with cyclical SGHMC alone and further to $q(h \star x) \le q(x)$2 with Wasserstein or $q(h \star x) \le q(x)$3 with MMD (Bendou et al., 21 Nov 2025).

A common misconception is that the method merely spreads parameters apart. The construction is more specific: it repels induced representation distributions, and its practical benefit is attributed to improved posterior coverage, reduced overfitting, and improved OOD generalization rather than to parameter diversity alone. The sensitivity curve in $q(h \star x) \le q(x)$4 is U-shaped, so excessive repulsion can overpower the likelihood and hurt convergence (Bendou et al., 21 Nov 2025).

3. Diffusion priors, blur attraction, and local sharpening in blind deconvolution

In blind deconvolution, the image formation model is

$q(h \star x) \le q(x)$5

with likelihood

$q(h \star x) \le q(x)$6

Using a diffusion prior $q(h \star x) \le q(x)$7, the negative log-posterior becomes

$q(h \star x) \le q(x)$8

where $q(h \star x) \le q(x)$9. The diffusion prior itself is described through a forward SDE, a reverse SDE involving the Stein score, and a probability flow ODE whose instantaneous change-of-variables formula evaluates $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$0 (Nguyen et al., 4 Aug 2025).

The central empirical result is that for natural images and several blur families,

$$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$1

so blurred images are more likely under the diffusion prior. This produces a global MAP pathology. If the kernel family contains $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$2 and the prior satisfies $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$3 for all $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$4 and $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$5, then the pair $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$6 is a global minimizer. The paper therefore concludes that the global MAP estimator tends to produce sharp filters close to the Dirac delta function and blurry solutions (Nguyen et al., 4 Aug 2025).

The sharpening mechanism appears not at the global level but at the level of local minima. Natural images lie near numerous second-order critical points of $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$7, with Hessian spectra showing many directions of large curvature transverse to the manifold and relatively few flat directions. Local intrinsic manifold dimensions are estimated as $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$8 for FFHQ-64 and $$\ker \nabla^2 q(\bar{x}) \cap \ker H_{\bar{\theta}} = \{0\}, \qquad \ker J(\bar{\theta}) = \{0\}, \qquad \big(H_{\bar{\theta}} \ker \nabla^2 q(\bar{x})\big) \cap \mathrm{Im}\,J(\bar{\theta}) = \{0\},$$9 for AFHQ-64. Under the identifiability conditions quoted above, $(\bar{x},\bar{\theta})$0 is a strict local minimizer and remains stable under small noise, with

$(\bar{x},\bar{\theta})$1

In this setting, the “repulsive” directions are precisely those excluded by the Hessian and injectivity conditions, which prevent collapse toward $(\bar{x},\bar{\theta})$2 in a neighborhood of the true solution (Nguyen et al., 4 Aug 2025).

The practical consequences follow directly. Motion and defocus kernels have Fourier transforms that vanish more slowly than Gaussian or Airy kernels, yielding sharper posterior profiles and making the local minimum around $(\bar{x},\bar{\theta})$3 more pronounced. Initializing $(\bar{x},\bar{\theta})$4 to a large blur and $(\bar{x},\bar{\theta})$5, alternating proximal image updates with single gradient steps on $(\bar{x},\bar{\theta})$6, and periodically resetting $(\bar{x},\bar{\theta})$7 to $(\bar{x},\bar{\theta})$8 enlarge the basin from which the favorable local minimum is reachable. Conversely, at larger noise levels the local minimum near $(\bar{x},\bar{\theta})$9 vanishes and the profile decreases toward small $\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$0, so the posterior becomes globally attractive to $\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$1 again (Nguyen et al., 4 Aug 2025).

This section also clarifies a broader misconception: sharpening does not imply that the entire posterior has become globally favorable to the desired sharp image. In this case, the global objective remains biased toward blur, while sharpening is local, conditional on initialization, parameterization, and noise level.

4. Cusp-like sharpening of tissue interfaces

In dense epithelial models, the underlying energy is

$\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$2

with overdamped Brownian dynamics

$\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$3

The interface is represented by a single-valued height field $\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$4, and the key surface-energy model is

$\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$5

The first term encodes the topological sharpening effect associated with coordinated four-fold vertices or very short edges along the interface (Yue et al., 2024).

The non-analyticity is the essential sharpened feature. For a displacement coordinate $\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$6, $\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$7 implies $\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$8, so even infinitesimal perturbations encounter a finite restoring force. In the coarse-grained interface description, the same mechanism acts through $\epsilon_s(y) = g_0 |h'(y)| + g_1 h'(y)^2.$9. By contrast, ordinary capillarity yields

$|h'|$0

with structure factor

$|h'|$1

For the cusp-only case,

$|h'|$2

the Brownian-bridge analysis gives

$|h'|$3

and with combined cusp plus harmonic terms,

$|h'|$4

(Yue et al., 2024).

The paper’s main quantitative point is that sharpening is scale dependent. In Voronoi simulations with $|h'|$5, the collapsed low-$|h'|$6 spectrum gives

$|h'|$7

At short scales, $|h'|$8 decreases from approximately $|h'|$9 to approximately $\mathrm{sign}(h')$0 as $\mathrm{sign}(h')$1 increases from $\mathrm{sign}(h')$2 to $\mathrm{sign}(h')$3, corresponding to $\mathrm{sign}(h')$4. The suppression correlates with the fraction of very short edges smaller than $\mathrm{sign}(h')$5 on the interface. By contrast, vertex simulations at the same parameters give

$\mathrm{sign}(h')$6

so no detectable sharpening appears at equilibrium in the vertex model at low $\mathrm{sign}(h')$7 (Yue et al., 2024).

The experimental comparison supports the same picture. In the confrontation assay of two epithelial monolayers, the spectrum follows capillary-wave theory before contact, but after a stable interface forms, strong high-$\mathrm{sign}(h')$8 suppression appears around $\mathrm{sign}(h')$9. Low-$$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$0 fits give $$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$1 before contact and $$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$2 after contact. The interpretation offered is that cell registration and four-fold vertices generate short-length-scale topological sharpening even when the long-wavelength effective tension changes only modestly (Yue et al., 2024).

Here the phrase “repulsive landscape” has a literal geometric meaning: the interface is locally repelled from nonzero slope by a cusp rather than by a quadratic penalty, and the resulting sharpening is strongest at short length scales rather than uniformly across the spectrum.

5. Tadpole charge and repulsive de Sitter directions

In Type IIB flux compactifications on the mirror of a rigid Calabi–Yau threefold, the scalar potential is

$$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$3

with

$$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$4

and the flux-induced D3 charge

$$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$5

For the orientifold action $$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$6, tadpole cancellation imposes

$$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$7

(Ishiguro et al., 2021).

The sharpening mechanism is encoded analytically in the axio-dilaton direction. Writing $$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$8, the potential contains

$$p(\theta \mid D) \propto p(D \mid \theta) p(\theta),$$9

so $\Phi$00 is the linear coefficient in $\Phi$01. Stable Minkowski or de Sitter extrema require a narrow window,

$\Phi$02

Because $\Phi$03 bounds $\Phi$04 by $\Phi$05, the paper argues that the allowed charge sector generically excludes stable de Sitter extrema (Ishiguro et al., 2021).

The numerical scans support this exclusion. In the isotropic scan over $\Phi$06 flux sets, there are $\Phi$07 SUSY AdS vacua, $\Phi$08 non-SUSY AdS stable vacua, and $\Phi$09 non-SUSY AdS unstable vacua, while SUSY or non-SUSY Minkowski vacua and unstable dS vacua were not found in that dataset. In the anisotropic scan over $\Phi$10 flux sets, $\Phi$11 SUSY AdS vacua, $\Phi$12 non-SUSY AdS stable vacua, and $\Phi$13 non-SUSY AdS unstable vacua were found, with no Minkowski or dS vacua. In a targeted isotropic scan, the only dS vacua within the tadpole-allowed sector are unstable, with explicit examples at $\Phi$14 satisfying the refined dS bound with $\Phi$15. By contrast, extending the search up to $\Phi$16 yields $\Phi$17 stable dS vacua and $\Phi$18 stable Minkowski vacuum, all in the swampland sector (Ishiguro et al., 2021).

In this setting, “repulsive” means that candidate dS critical points within the allowed flux-charge window are not metastable: they exhibit either a gradient satisfying $\Phi$19 or a tachyonic direction satisfying $\Phi$20. The landscape is therefore sharpened by a consistency condition rather than by an added interaction term: tadpole cancellation structurally excises the stable dS sector from the landscape (Ishiguro et al., 2021).

6. Topography transitions in repulsive binary mixtures

For classical symmetric binary mixtures of point particles with purely repulsive pair potentials, the total energy is

$\Phi$21

with control parameter $\Phi$22 and equal composition $\Phi$23. The quench protocol starts from an ideal-gas configuration at $\Phi$24 and performs conjugate-gradient minimization to the nearest inherent structure. The paper studies Uhlenbeck–Ford, inverse-power-law, WCA, and Gaussian-core interactions and reports two distinct ordering outcomes controlled by the unlike-particle repulsion (Santos-Flórez et al., 2020).

For strong A–B repulsion, the inherent structure shows chemical ordering via unmixing, with morphology resembling spinodal decomposition and mean particle displacement much larger than the mean spacing $\Phi$25. For weak A–B repulsion, the inherent structure is a polycrystalline rock-salt solid at homogeneous composition, and the descent is barrierless. The mean particle displacement then scales as $\Phi$26. In the Uhlenbeck–Ford case, the approximate crystallization window is bounded by $\Phi$27 and $\Phi$28, while the unmixing transition occurs at $\Phi$29 (Santos-Flórez et al., 2020).

The paper interprets this as a transition in the topography of the potential-energy landscape. As $\Phi$30 decreases, AB contacts become energetically less costly relative to AA and BB, and the landscape develops prominent funnels guiding descent to B1 polycrystals. As $\Phi$31 increases, crystalline basins lose dominance, amorphous basins dominate at intermediate $\Phi$32, and sufficiently strong $\Phi$33 favors demixed basins. The breadth of the crystalline window depends on the behavior of the potential near $\Phi$34: moving from logarithmic UF to IPL4 to IPL6 to WCA, the crystallization window shrinks systematically, while in the Gaussian-core case it is extremely narrow and disappears for $\Phi$35 (Santos-Flórez et al., 2020).

The practical control parameter is the contact-repulsion ratio

$\Phi$36

which reduces to $\Phi$37 in the prefactor sense for diverging potentials and equals $\Phi$38 in the Gaussian-core case. Empirically, small $\Phi$39 correlates with crystalline inherent structures, whereas large $\Phi$40 correlates with unmixed inherent structures. The structural diagnostics include $\Phi$41, $\Phi$42, $\Phi$43, and the grain-size distribution

$\Phi$44

for the largest grains in a crystalline Uhlenbeck–Ford sample with $\Phi$45 (Santos-Flórez et al., 2020).

Here sharpening means a reorganization of basin depth, steepness, and catchment volume. The repulsive interaction does not simply stiffen the system uniformly; rather, it controls whether the descent is captured by crystalline funnels, amorphous metastable states, or demixed minima.

7. Comparative interpretation and recurrent misconceptions

Across these domains, a plausible synthesis is that repulsive landscape sharpening acts by restructuring accessibility rather than by uniformly increasing curvature everywhere. In ReBaPL, representation-space repulsion plus cyclical SGHMC separates posterior modes and prevents premature collapse to a single prompt solution (Bendou et al., 21 Nov 2025). In blind deconvolution, the global posterior remains attractive to blur, but local Hessian structure and identifiability create sharply stable minima near realistic sharp images (Nguyen et al., 4 Aug 2025). In epithelial interfaces, cusp-like terms suppress short-scale fluctuations far more strongly than harmonic capillarity, but the effect is explicitly scale dependent (Yue et al., 2024). In flux compactifications, tadpole cancellation narrows the stability window so strongly that candidate dS extrema within the allowed sector become repulsive in the sense of refined swampland bounds (Ishiguro et al., 2021). In repulsive mixtures, tuning unlike-particle repulsion reorganizes the potential-energy landscape between crystalline, amorphous, and demixed funnels (Santos-Flórez et al., 2020).

Several misconceptions are therefore excluded by the evidence. First, sharpening is not equivalent to global convexification: the diffusion-prior case shows the opposite, because global MAP can still favor the no-blur solution. Second, repulsion is not necessarily parameter-space exclusion: in ReBaPL it is computed on induced representation distributions. Third, sharpening is not necessarily scale independent: tissue interfaces exhibit modest low-$\Phi$46 renormalization but strong high-$\Phi$47 suppression. Fourth, stronger repulsion is not uniformly beneficial: ReBaPL reports a U-shaped dependence on repulsion strength, blind deconvolution loses the favorable local minimum at higher noise, and repulsive mixtures pass from crystallization to unmixing as $\Phi$48 increases (Bendou et al., 21 Nov 2025, Nguyen et al., 4 Aug 2025, Yue et al., 2024, Santos-Flórez et al., 2020).

A final implication is methodological. The cited works suggest three distinct routes to sharpened landscapes: adding explicit repulsive energies between particles or modes, exploiting local curvature and identifiability to create favorable basins in an otherwise biased objective, and imposing consistency constraints that remove metastable sectors altogether. The open problems listed in the sources follow directly from these mechanisms: convergence theory for cyclical SGHMC with repulsive potentials, prior design that breaks the blur-attraction inequality while preserving diffusion quality, adaptive metrics and schedules for representation-space repulsion, and broader characterization of how geometric constraints or Hessian structure can enforce repulsion away from degenerate directions (Bendou et al., 21 Nov 2025, Nguyen et al., 4 Aug 2025).