Curvature-Aware Structural Enhancement (CASE)

• CASE is a geometric strategy that employs curvature as an explicit structural prior to delineate, encode, and enhance complex structures across various data modalities.
• It utilizes a range of curvature measures—from squared and absolute curvature to Ricci and effective resistance—to guide sub-pixel structure estimation, segmentation, and trajectory shaping.
• CASE leverages advanced optimization and non-local encoding techniques, resulting in improved fidelity and stability in applications such as image enhancement, graph embedding, and language model pruning.

Curvature-Aware Structural Enhancement (“CASE”, Editor’s term) denotes a class of methods in which curvature, or a curvature surrogate, is used as an explicit structural prior for enhancement, delineation, encoding, trajectory shaping, or pruning. Across the supplied literature, CASE appears in several technically distinct forms: curvature regularization of center-lines and surfaces for sub-pixel thin-structure estimation, squared-curvature energies for segmentation and inpainting, curvature-prior elastica geodesics for curvilinear tracking, rank-informed constraints that “straighten” reverse diffusion trajectories, discrete Ricci- or resistance-based graph encodings and message-passing weights, heterogeneous manifolds whose scalar curvature matches node-wise graph curvature, and Fisher-normalized kinetic criteria that isolate reasoning-relevant structural pathways in LLMs (Marin et al., 2015, Hou et al., 2023, Fesser et al., 2023, Qian, 9 May 2026, Chen et al., 2023).

1. Genealogy and conceptual scope

The earliest supplied works place CASE in variational vision. “Efficient Regularization of Squared Curvature” formulates squared-curvature regularization through integral geometry and straight line triple cliques, emphasizing preservation of elongated structures and fine details while avoiding strong block artifacts at high angular resolution (Nieuwenhuis et al., 2013). “Thin Structure Estimation with Curvature Regularization” then moves from boundary regularization to simultaneous detection and delineation of thin structures with sub-pixel localization and real-valued orientation estimation, using curvature as a prior on center-lines or surfaces rather than only on explicit contours (Marin et al., 2015).

Subsequent work broadens the role of curvature far beyond classical image regularization. In graph representation learning, “Heterogeneous manifolds for curvature-aware graph embedding” introduces a single extra radial dimension so that scalar curvature varies with position and can match node-wise Forman curvature, while “Effective Structural Encodings via Local Curvature Profiles” defines a 5-dimensional node encoding from Ollivier–Ricci edge curvatures and shows that curvature-based structural encodings can outperform rewiring (Giovanni et al., 2022, Fesser et al., 2023). “Efficient Curvature-aware Graph Network” replaces Ollivier–Ricci curvature with Effective Resistance Curvature as a cheaper but geometrically consistent surrogate for message passing and pooling (Fei et al., 3 Nov 2025).

The same pattern also appears in generative modeling and LLMs. “Global Structure-Aware Diffusion Process for Low-Light Image Enhancement” interprets the reverse diffusion process as an ODE trajectory and constrains it through rank-informed non-local structure matching, presenting this as curvature-aware trajectory regularization rather than as a direct penalty on second derivatives (Hou et al., 2023). “Relative Kinetic Utility for Reasoning-Aware Structural Pruning in LLMs” treats pruning as a geometry-aware problem over model depth, where Fisher trace normalization acts as a lightweight curvature-aware normalization that preserves “kinetic spikes” and high-curvature logical routing (Qian, 9 May 2026). This suggests that CASE is not tied to a single data modality; it is a recurring design principle in which curvature organizes structural decisions.

2. Geometric primitives and recurring mathematical forms

In thin-structure estimation, the core variables are binary indicators $X=\{x_i\}$ and geometric variables $L=\{l_i\}$, with curvature discretized over neighboring tangents. For squared curvature, the base energy is

$$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$

while absolute curvature is approximated by

$$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$

The paper distinguishes the two regimes sharply: squared curvature encourages smooth curves, whereas absolute curvature prefers piecewise-straight curves with a small number of sharp corners (Marin et al., 2015). In the related squared-curvature regularizer of (Nieuwenhuis et al., 2013), the continuous functional

$K(S)= \int_{\partial S} \kappa^2 \, ds$

is approximated by weighted counts of triple-clique configurations, with weights

$w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$

This converts curvature into a quadratic pseudo-Boolean energy whose pairwise terms decompose into submodular and supermodular components (Nieuwenhuis et al., 2013).

In curvature-prior geodesics, the preferred quantity is not $\kappa^2$ but $(\kappa-\omega)^2$, where $\omega:\mathcal{M}\to\mathbb{R}$ is a data-driven signed curvature prior defined on the lifted orientation space $\mathcal{M}=\Omega\times S^1$. The associated metric is

$L=\{l_i\}$0

Its geometric effect is explicit: the energetically cheap controls are tilted toward local curvature $L=\{l_i\}$1 instead of toward zero curvature, so highly bent but structurally plausible continuations become cheap when supported by the prior (Chen et al., 2023).

Graph methods use different curvature objects. Local Curvature Profiles first compute Ollivier–Ricci curvature

$L=\{l_i\}$2

on edges, then summarize the incident curvature multiset at a node by

$L=\{l_i\}$3

ERC replaces optimal transport with effective resistance, defining

$L=\{l_i\}$4

where $L=\{l_i\}$5 is effective resistance. The intended interpretation remains Ricci-like: positively curved edges lie in dense, redundant regions; negative curvature marks bottlenecks (Fesser et al., 2023, Fei et al., 3 Nov 2025).

In graph embedding, curvature is neither pairwise nor pathwise but scalar and positional. A rotationally symmetric metric

$L=\{l_i\}$6

is added to a homogeneous base manifold, yielding a product space with scalar curvature

$L=\{l_i\}$7

Because $L=\{l_i\}$8 is monotone in $L=\{l_i\}$9, the extra radial coordinate becomes a one-dimensional control for matching node-wise Forman curvature (Giovanni et al., 2022).

In diffusion and LLM pruning, curvature appears through surrogates. The low-light diffusion paper regularizes the reverse trajectory by matching singular-value spectra of non-local patch-group matrices: $$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$0 with a schedule

$$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$1

The paper states that this acts as a curvature-aware constraint on the ODE trajectory (Hou et al., 2023). RKU, by contrast, uses a continuous kinetic integral over model depth and Fisher-normalized utility,

$$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$2

where the normalized utility is interpreted as curvature-aware in the natural-gradient sense (Qian, 9 May 2026).

3. Computational realizations

The optimization strategies behind CASE are heterogeneous, but they share a reluctance to rely on purely local or purely first-order decisions. In thin-structure estimation, naive block-coordinate descent is rejected because tangents in background regions remain unconstrained and disconnected fragments cannot “see” that they should be connected. The alternative is a mean-field variational lower-bound scheme: an $$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$3-step solves a non-linear least-squares problem for tangents using an inexact Levenberg–Marquardt trust-region method, and a $$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$4-step updates Bernoulli probabilities $$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$5 with a logistic mean-field rule (Marin et al., 2015).

The squared-curvature clique model in (Nieuwenhuis et al., 2013) converts curvature regularization into pairwise submodular and supermodular terms and minimizes the resulting mixed energy with the trust region framework, specifically LSA-TR. The main technical point is that high angular resolutions remain feasible because the number of cliques grows linearly with the number of orientations rather than combinatorially.

Curvature-prior geodesics use a different machinery entirely. The weighted metric $$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$6 induces a static Hamilton–Jacobi–Bellman equation,

$$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$7

discretized by an adaptive upwind finite-difference scheme on $$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$8 and solved in a single pass with a generalized Fast-Marching method. Backtracking then integrates the geodesic ODE

$$E(L,X)= \sum_{(i,j)\in N} \kappa^2(l_i,l_j)\, x_i x_j \;+\; \sum_{i} \frac{1}{\sigma^2}\,\|l_i - \tilde p_i\|_+^2\, x_i \;+\; \sum_i \lambda_i x_i,$$9

from target to source, yielding a globally optimal curve with fixed endpoints and tangents (Chen et al., 2023).

The diffusion formulation mixes standard DDPM training with two extra structures. First, an uncertainty branch predicts $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$0 and is pre-trained with

$$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$1

Second, the main training objective combines uncertainty-weighted noise estimation and scheduled rank-based structure regularization. No special non-local module is inserted into the U-Net; the structure-awareness is imposed externally through patch clustering, SVD, and the loss (Hou et al., 2023).

Graph methods typically precompute curvature once and inject it into otherwise standard architectures. LCP is concatenated to node features and fed unchanged into GCN, GIN, or GAT, whereas ERC modifies message passing through a curvature-weighted operator $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$2 or rewrites graph weights for CurvPool as

$$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$3

The computational gap between the two graph-curvature regimes is central: ORC requires optimal transport, while ERC uses Laplacian inversion or sparse linear solves after a perturbation $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$4 (Fesser et al., 2023, Fei et al., 3 Nov 2025).

RKU operationalizes curvature-aware pruning as one-shot structural pruning. It collects activation–gradient products along depth for a continuous objective $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$5, integrates these signals over layers, normalizes them per layer, forms $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$6, and then prunes the lowest-scoring FFN channels without heavy retraining. Short LoRA PEFT appears only as a probe of post-pruning plasticity, not as a prerequisite for the reported sparsity results (Qian, 9 May 2026).

4. Representative instantiations across domains

The supplied literature supports a domain-spanning but non-uniform view of CASE.

Domain	Structural object	Curvature mechanism
Thin-structure estimation	Center-lines or surfaces with sub-pixel location and real-valued orientation	Absolute or squared curvature over neighboring tangents (Marin et al., 2015)
Segmentation and inpainting	Binary boundaries on a grid	Squared-curvature integral approximated by weighted straight triple cliques (Nieuwenhuis et al., 2013)
Curvilinear geodesics	Curves with fixed endpoints and endpoint tangents	Elastica metric with preferred curvature $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$7 (Chen et al., 2023)
Low-light diffusion	Reverse denoising trajectory in image space	Rank-informed structural loss acting as curvature-aware ODE regularization (Hou et al., 2023)
GNN structure modeling	Node encodings, message passing, and pooling	LCP from ORC; ERC-based reweighting and CurvPool (Fesser et al., 2023, Fei et al., 3 Nov 2025)
Graph embedding and LLM pruning	Node coordinates; FFN channels	Radial scalar-curvature matching; Fisher-normalized kinetic utility (Giovanni et al., 2022, Qian, 9 May 2026)

In vision, CASE centers on coherent recovery of curves, surfaces, and boundaries. Curvature regularization links weak fragments, suppresses isolated false positives, and moves detected structure off the pixel grid by projecting observations onto tangent lines or planes. When the prior is data-driven and signed, as in elastica with $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$8, the same principle becomes directional and branch-sensitive rather than merely smoothness-promoting (Marin et al., 2015, Chen et al., 2023).

In generative enhancement, CASE is not a post hoc contour regularizer but a trajectory-shaping mechanism. The diffusion paper explicitly frames low-curvature ODE trajectories as more stable and effective, then anchors intermediate samples to ground-truth structural spectra in non-local patch space. This is curvature-aware structural enhancement in the strict sense that the “structure” being preserved is non-local and rank-informed rather than purely edge-based (Hou et al., 2023).

In graph learning, CASE splits into at least three variants. Feature-level enhancement appears in LCP, topology-sensitive weighting and pooling appear in ERC-based GNNs, and geometric representation enhancement appears in heterogeneous manifolds whose scalar curvature varies by position. The LCP work is particularly explicit that local curvature encodings and global positional encodings are complementary, while the ERC work emphasizes scalability and substitutability for ORC (Fesser et al., 2023, Fei et al., 3 Nov 2025, Giovanni et al., 2022).

In LLMs, the structural object is neither a curve nor a graph edge but a reasoning pathway through depth. The RKU paper treats high-gradient, moderate-activation channels as “kinetic spikes” that carry high-curvature logical routing, and uses Fisher trace normalization to prevent magnitude-based heuristics from preserving only high-frequency syntactic channels (Qian, 9 May 2026). A plausible implication is that CASE can be understood as a geometry-aware criterion for deciding which structures should be preserved, suppressed, or amplified, even when those structures are latent.

5. Empirical behavior and comparative findings

The vision papers report that curvature-aware models improve both fidelity and structure. For 2D edges, “Thin Structure Estimation with Curvature Regularization” reports an F-measure around 0.83 versus 0.84 for the best specialized edge detector, while also producing smoother, better localized, sub-pixel edges; in 3D, it demonstrates vessel centerline estimation on a $$|\kappa(l_i,l_j)| = \frac{\|l_i-p_j\| + \|l_j-p_i\|}{\|p_i-p_j\|}.$$9 CT volume using pruning and subsampling strategies (Marin et al., 2015). “Efficient Regularization of Squared Curvature” reports accurate and visually pleasing solutions without strong artifacts at reasonable run times, and emphasizes that squared curvature preserves limbs and thin details that length regularization removes (Nieuwenhuis et al., 2013). For curvature-prior geodesics, the quantitative contrast is sharper: average Jaccard scores of $K(S)= \int_{\partial S} \kappa^2 \, ds$0 and $K(S)= \int_{\partial S} \kappa^2 \, ds$1 for the curvature-prior model versus $K(S)= \int_{\partial S} \kappa^2 \, ds$2 and $K(S)= \int_{\partial S} \kappa^2 \, ds$3 for classical elastica under the two tested $K(S)= \int_{\partial S} \kappa^2 \, ds$4 values (Chen et al., 2023).

The low-light diffusion results present the same structural theme in a different metric space. On LOLv1, the method reports PSNR $K(S)= \int_{\partial S} \kappa^2 \, ds$5 dB, SSIM $K(S)= \int_{\partial S} \kappa^2 \, ds$6, and LPIPS $K(S)= \int_{\partial S} \kappa^2 \, ds$7, compared with SNR-Aware at $K(S)= \int_{\partial S} \kappa^2 \, ds$8 dB, $K(S)= \int_{\partial S} \kappa^2 \, ds$9, and $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$0. In the regularization ablation, the baseline $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$1 rises to $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$2 with rank-based modeling, and the full combination of rank-based regularization, schedule, and uncertainty reaches $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$3 (Hou et al., 2023).

Graph results indicate that curvature can act as a stronger structural encoding than either local degree profiles or curvature-based rewiring. With GCN on IMDB, LCP achieves $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$4 versus $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$5 with no encoding and $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$6 with Laplacian eigenvectors; with GCN on MUTAG it reaches $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$7, and on ENZYMES $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$8. The paper also states that utilizing curvature information for structural encodings delivers significantly larger performance increases than rewiring (Fesser et al., 2023). ERC-based GNNs preserve most of the predictive gains of ORC while sharply reducing overhead: reported speedups range from $w_i = \frac{4\cdot\Delta\theta_i}{d_i^3}.$9 to $\kappa^2$0, with Amazon Computers taking $\kappa^2$1 s for ORC on CPU2 versus $\kappa^2$2 s for ERC on GPU L20 (Fei et al., 3 Nov 2025).

Curvature-aware graph embedding yields improvements on higher-order structure rather than merely on pairwise distortion. On triangle-count distortion, $\kappa^2$3 improves Aves-Wildbird from $\kappa^2$4 to $\kappa^2$5, WebEdu from $\kappa^2$6 to $\kappa^2$7, and Facebook from $\kappa^2$8 to $\kappa^2$9. The same work shows that heterogeneous manifolds can produce dense communities while preserving heavy-tailed degree behavior more effectively than simply increasing the global distance threshold in a homogeneous hyperbolic space (Giovanni et al., 2022).

The LLM pruning results show the same “curvature protects structure” pattern under high sparsity. On Qwen-2.5-7B at $(\kappa-\omega)^2$0 sparsity without fine-tuning, RKU reaches $(\kappa-\omega)^2$1 on GSM8K and $(\kappa-\omega)^2$2 on AQuA, compared with $(\kappa-\omega)^2$3 and $(\kappa-\omega)^2$4 for Wanda-Struct and $(\kappa-\omega)^2$5 and $(\kappa-\omega)^2$6 for Taylor-FO. The paper also reports that raw AGF without Fisher normalization can collapse to $(\kappa-\omega)^2$7 accuracy on AQuA in some settings, and that RKU better preserves heavy-tailed singular-value spectra in deep-layer activations (Qian, 9 May 2026). This empirical pattern recurs throughout CASE: curvature-aware structure selection tends to matter most when ambiguity, sparsity, or long-range dependence would otherwise make local heuristics brittle.

6. Limitations, misconceptions, and research directions

A first misconception is that CASE refers to one curvature definition. The supplied papers use at least six distinct notions: absolute or squared center-line curvature, squared-curvature integrals on binary boundaries, preferred curvature fields $(\kappa-\omega)^2$8, Ollivier–Ricci curvature, Effective Resistance Curvature, scalar curvature on heterogeneous manifolds, and Fisher/Hessian proxies in depth-integrated pruning (Marin et al., 2015, Nieuwenhuis et al., 2013, Chen et al., 2023, Fesser et al., 2023, Fei et al., 3 Nov 2025, Giovanni et al., 2022, Qian, 9 May 2026). A second misconception is that curvature-awareness must be a direct second-derivative penalty. The diffusion paper implements it indirectly through rank-based structure matching and time-varying anchoring, while RKU implements it through Fisher trace normalization of kinetic utility (Hou et al., 2023, Qian, 9 May 2026).

The main practical constraints are computational and statistical. Thin-structure estimation reports runtimes of about $(\kappa-\omega)^2$9 s for a 2D edge example and about one day for a large 3D vessel example on CPU, although the operations are described as highly parallelizable and suitable for GPU (Marin et al., 2015). Exact ORC is expensive; LCP explicitly notes $\omega:\mathcal{M}\to\mathbb{R}$0 complexity for ORC-LCP, and ERC is motivated by the same bottleneck (Fesser et al., 2023, Fei et al., 3 Nov 2025). Diffusion adds overhead from patch extraction, clustering, and SVD during training, and remains slower than LLFlow, SNR-Aware, and LLFormer, with $\omega:\mathcal{M}\to\mathbb{R}$1 s versus $\omega:\mathcal{M}\to\mathbb{R}$2–$\omega:\mathcal{M}\to\mathbb{R}$3 s per image on LOLv1 with RTX3080 (Hou et al., 2023).

The methods also inherit domain-specific assumptions. Thin-structure energies are pairwise over neighboring sites and do not explicitly model complex junction geometry; further interpolation is needed for continuous curve reconstruction (Marin et al., 2015). The elastica-prior framework depends on reliable segmentation and skeletonization, removes junctions before constructing the prior, and is implemented only in 2D (Chen et al., 2023). LCP gains are smaller on some node-classification tasks, and exact ORC remains expensive; ERC can differ from ORC in distributional shape even when downstream accuracy is similar (Fesser et al., 2023, Fei et al., 3 Nov 2025). Heterogeneous graph embeddings currently assume undirected graphs and a fixed warping function $\omega:\mathcal{M}\to\mathbb{R}$4 rather than a learned curvature profile (Giovanni et al., 2022). RKU deliberately trades some static retrieval performance for deep reasoning, and its behavior depends on the reasoning relevance of the data used to estimate AGF (Qian, 9 May 2026).

The open directions described in the supplied works are consistent with a broader CASE agenda. Thin-structure regularization is proposed as a differentiable post-processing layer or loss component for deep models, with possible anisotropic or data-dependent curvature and improved curvature approximations (Marin et al., 2015). The diffusion work suggests that structure-aware trajectory shaping may extend beyond low-light enhancement and that extremely low-light patches may need different treatment (Hou et al., 2023). The graph papers point toward multi-hop curvature profiles, learnable curvature transformations, curvature-aware attention, cheaper substitutes for ORC, and adaptive combinations of encodings and rewiring (Fesser et al., 2023, Fei et al., 3 Nov 2025). The heterogeneous-manifold paper explicitly raises learning $\omega:\mathcal{M}\to\mathbb{R}$5, moving beyond scalar curvature, and extending to directed graphs (Giovanni et al., 2022). The pruning paper suggests growth mechanisms, RL-based reasoning settings, and layer-specific structural redesign based on curvature-aware scores (Qian, 9 May 2026). Taken together, these directions suggest that CASE is best viewed not as a fixed algorithmic family but as a transferable geometric strategy: estimate or approximate structure-relevant curvature, couple it to an optimization or learning objective, and use it to preserve coherent pathways that local evidence alone would underspecify.