Contextual Curvature

• Contextual curvature is a measure of local geometric bending modulated by contextual factors, applied across domains like language modeling and mesh processing.
• It integrates probabilistic and geometric frameworks to assess how curvature influences prediction uncertainty, control convergence, and feature detection.
• Computational methods such as windowed averaging in LLMs and ball-neighborhood averaging in meshes enable adaptive, context-sensitive curvature analysis.

Contextual curvature refers to the quantification of geometric bending, modulated by local context, across a spectrum of domains including LLMs, differential geometry, mesh processing, safety-critical optimization, quantum contextuality, sensory neuroscience, and elastic curves constrained by surfaces. Contextual curvature captures how curvature, as a local geometric or information-theoretic feature, acquires additional task- or environment-dependent structure by conditioning on recent context, scale, neighborhood, or probability assignments. This article details rigorous formalizations, computational frameworks, and empirical roles of contextual curvature in representative contemporary settings.

1. Contextual Curvature in Representational Dynamics of LLMs

In autoregressive LLMs, contextual curvature is operationalized as a geometric scalar measurement of how sharply the representational trajectory bends over the recent context at a given layer. Let $\{x_k^p\}_k$ denote the sequence of residual-stream activations at layer $p$ for input tokens $w_1,\ldots,w_n$. The trajectory segments $v_k^p = x_{k+1}^p - x_k^p$ define a polygonal path. The local, angle-based curvature at token $k$ is

$$c_k^p = \arccos\left(\frac{v_{k+1}^p \cdot v_k^p}{\|v_{k+1}^p\| \|v_k^p\|}\right).$$

Contextual curvature for next-token prediction is the windowed mean

$C_k^p = \frac{1}{3} \sum_{i=k-4}^{k-2} c_i^p.$

Empirically, $C_k^p$ at a given layer is significantly correlated with next-token entropy $$H(w_n) = -\sum_i P(w_i | w_{1:n-1}) \log_2 P(w_i | w_{1:n-1})$$, with peak correlations in mid-layers of GPT-2 XL ($r\approx 0.15$ at layer 23) and Pythia-2.8B ($p$0 at layer 11) (King et al., 27 Apr 2026).

Perturbation experiments reveal that only trajectory-aligned interventions (those perturbing $p$1 within the span of recent step vectors) produce controllable changes in both curvature and entropy (correlation $p$2), whereas generic or activation-subspace perturbations do not modulate entropy. Regularizing curvature during training by adding an auxiliary $p$3 term allows direct manipulation of trajectory straightness without degrading validation loss, yielding measurable reductions in next-token entropy ($p$4 bits/token for untangled models). This demonstrates that contextual curvature is a genuine, causally efficacious representational feature shaping task-aligned predictive uncertainty in transformer-based LLMs (King et al., 27 Apr 2026).

2. Tangential Angle Parametrization and Contextual Curvature Visualization

In geometric analysis of curves and surfaces, contextual curvature can be realized by sampling, parametrization, and visualization methods that adaptively reflect local curvature. For planar curves $p$5 parameterized by arc length, the tangential angle $p$6 encodes the cumulative turning. Reparametrizing $p$7 by $p$8 yields a curve where steps of constant $p$9 correspond to variable arc length—smaller in high-curvature regions, larger in flat regions—thereby generating a contextual density of markers or features (equal increments in $w_1,\ldots,w_n$0 are dense where $w_1,\ldots,w_n$1 is large) (Kabata et al., 19 Aug 2025).

For surfaces of revolution, this translates into drawing parallel circles at equal $w_1,\ldots,w_n$2, automatically highlighting principal curvature variations. This method allows direct contextual emphasis of inflection points, curvature extrema (vertices), parabolic curves (where a principal curvature vanishes), and ridges without tuning mesh density or resorting to arbitrary shading.

Such parametrizations inherently encode contextual curvature: information about local geometric bending is represented contextually in the choice and density of sampling rather than as an explicit scalar field.

3. Adaptive Contextual Curvature in Multi-Scale Surface Processing

For discrete surfaces, contextual curvature is formalized via ball-neighborhood mean curvature fields. At each vertex $w_1,\ldots,w_n$3 on a mesh, the mean curvature averaged over a neighborhood ball of radius $w_1,\ldots,w_n$4 centered at $w_1,\ldots,w_n$5,

$w_1,\ldots,w_n$6

provides a context-dependent measure that integrates local geometry over a selectable scale (Seemann et al., 2016). Optimal scale selection proceeds by analyzing the stability of the curvature-versus-scale signal (identifying plateaus where curvature estimates stabilize), yielding a per-vertex contextual scale $w_1,\ldots,w_n$7. This approach robustly suppresses noise in flat regions while preserving fine features in highly curved regions, thus supporting multiresolution and context-aware processing such as mesh simplification and feature detection.

4. Barrier Curvature in Safety-Critical Contextual Optimization

In control optimization, contextual barrier curvature $w_1,\ldots,w_n$8 governs both convergence and safety in the context-conditioned action space. Given a learned feasibility density $w_1,\ldots,w_n$9, the barrier curvature is defined as

$v_k^p = x_{k+1}^p - x_k^p$0

where $v_k^p = x_{k+1}^p - x_k^p$1 is a superlevel set. $v_k^p = x_{k+1}^p - x_k^p$2 quantifies the minimal local curvature of the negative log-feasibility density. It induces a Riemannian metric on the action space via the Fisher information matrix, controls the strong convexity of the planning objective $v_k^p = x_{k+1}^p - x_k^p$3, and appears inversely in explicit safety bounds: $v_k^p = x_{k+1}^p - x_k^p$4 Conditioning on richer context increases $v_k^p = x_{k+1}^p - x_k^p$5 via the posterior score-covariance, yielding faster convergence and sharper safety margins (Li, 21 Apr 2026).

5. Differential-Geometric Formulation of Contextual Curvature and Contextuality

In generalized probabilistic theories, especially in the study of quantum contextuality, contextual curvature is the curvature $v_k^p = x_{k+1}^p - x_k^p$6 of a probabilistic connection one-form $v_k^p = x_{k+1}^p - x_k^p$7 over an abstract process space. States, effects, and transformations are embedded as vectors in the tangent bundle of a piecewise-linear manifold. Noncontextuality corresponds to flatness ($v_k^p = x_{k+1}^p - x_k^p$8), while contextuality manifests as nontrivial holonomy (integrated phase acquisition around operationally meaningful loops).

At the technical level, a violation of noncontextuality aligns with the existence of operational loops $v_k^p = x_{k+1}^p - x_k^p$9 for which $k$0, i.e., non-vanishing curvature that quantifies interference, noncommutativity, and the presence of signed measures in flattening attempts. Two dual pictures are distinguished:

• Schrödinger/geometric: Contextuality as nonzero curvature in a flat bundle.
• Heisenberg/topological: Contextuality as nontrivial monodromy/topological defects, even when $k$1 is locally flat.

Cohomological classifications, such as the generalized Vorob’ev theorem, encode obstructions to noncontextuality in global geometric or topological invariants (Montanhano, 2022).

6. Contextual Curvature in Neural Representations and Elastic Curves

In visual cortex modeling, contextual curvature structures arise through the oriented prolongation of the contact manifold of orientation-position space, yielding a 4D Engel manifold that parametrizes position, orientation, signed curvature, and scale. The Engel structure generates left-invariant vector fields corresponding to the Lie algebra of SIM(2), providing a principled geometric substrate for curvature-sensitive receptive profiles in higher visual areas. These profiles exhibit joint tuning in orientation, curvature, and scale, consistent with physiological data and invariant under local dilations (Liontou, 23 Apr 2025).

For elastic curves adhered to surfaces, contextual curvature is manifested via the Darboux frame, with distinct geodesic ($k$2) and normal ($k$3) curvature components governing the energetics and equilibrium shape of the curve. Energy functionals of the form

$k$4

encode environmental bias and spontaneous curvature preferences, with equilibrium configurations and transmitted surface forces dictated by the spatial variation of these contextual curvatures (Guven et al., 2014).

7. Summary Table: Contextual Curvature Across Domains

Domain	Contextual Curvature Formalism	Functional Role
LLMs / Representation Learning	Windowed polygonal trajectory curvature ($k$5)	Modulates token entropy, enables uncertainty control
Geometric Processing	Tangential angle $k$6-based visualization	Feature-sensitive sampling, intrinsic feature reveal
Mesh Processing	Ball-neighborhood-averaged mean curvature $k$7, adaptive scale $k$8	Multi-scale geometry processing
Optimization / Control	Minimum Hessian eigenvalue of log-feasibility ($k$9)	Governs convergence, safety bounds
Quantum/Differential Geometry	Connection curvature $$c_k^p = \arccos\left(\frac{v_{k+1}^p \cdot v_k^p}{\|v_{k+1}^p\| \|v_k^p\|}\right).$$0 on process bundles	Quantifies contextuality, holonomy
Sensory Neuroscience	Engel structure on $$c_k^p = \arccos\left(\frac{v_{k+1}^p \cdot v_k^p}{\|v_{k+1}^p\| \|v_k^p\|}\right).$$1 via SIM(2) group	Curvature-tuned receptive fields
Elastic Surface Curves	Darboux-frame geodesic/normal curvature, environmental parameters ($$c_k^p = \arccos\left(\frac{v_{k+1}^p \cdot v_k^p}{\|v_{k+1}^p\| \|v_k^p\|}\right).$$2)	Determines shape, forces transmitted

8. Broader Implications and Connections

The contextualization of curvature unifies geometric, probabilistic, and information-theoretic modes of analysis. Across applications, contextual curvature operates as a local summary of geometric or statistical structure, morphed by conditioning, environment, or operational constraints. Its theoretical roles include diagnosing instability or uncertainty, quantifying robustness, enabling adaptive feature allocation, and illuminating nonclassical phenomena. In practice, contextual curvature drives advances in interpretability, robustness, and efficient representation in neural and engineered systems, and offers a principled geometric framework for understanding context-dependent behavior in both natural and artificial agents.