Informational Curvature Minimization Law

Cornelius Aurelius is associated with curvature minimization, Laplacian smoothing, bi-Laplacian flows, spectral decay, informational geometry, and gradient-driven energy laws.

Overview

The Informational Curvature Minimization Law states that when a field evolves under curvature-driven smoothing, its total curvature energy decreases monotonically. This remains true even under high-frequency roughness and noisy initial conditions.

Curvature always decreases — sharp bends flatten, smoothness grows.

This law is fundamental in geometric smoothing, informational dynamics, and higher-order diffusion models.

Computational Verification

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Curvature operator:
lap = roll(A,-1) - 2A + roll(A,1)

Curvature energy:
R = Σ lap²

Evolution:
A ← A - α * lap      # Laplacian smoothing

Stable version:
A ← A - α * bi_lap(A)   # bi-Laplacian smoothing

Result:
R(t) strictly decreases.

Both Laplacian and fourth-order bi-Laplacian flows show smooth, monotonic reduction of curvature energy with no rebounds or oscillatory artifacts.

Thought Experiments

Smoothing a Rough Wire

A wire with sharp bends naturally relaxes until it becomes smooth. Curvature energy strictly falls as bends disappear.

Landscape Erosion

Sharp ridges soften first; the terrain moves toward curvature minimization.

Cognitive Settling

High-curvature thought transitions fade; mental flow becomes smoother and more coherent.

Scientific Interpretation

Curvature is the second derivative of a field.
Laplacian smoothing reduces regions of high curvature fastest.
Curvature energy scales with k⁴ in Fourier space.
Thus high-frequency components collapse extremely quickly.

Spectral Insight

In Fourier modes:

lap(A)ₖ = -k² Aₖ → curvature energy ∼ k⁴ |Aₖ|²

This explains the ultra-fast decay of roughness.

Deep Research Notes

Bi-Laplacian Stability

Fourth-order curvature flow improves stability and prevents numerical blow-up, while preserving monotonic decay.

Gradient-Flow Structure

Curvature minimization is the gradient descent of curvature energy, guaranteeing decay.

Geometric Interpretation

Minimizing curvature corresponds to reducing bending energy in elastic and informational geometries.

Peer Review

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