This law describes how an informational field subjected to fourth-order diffusion (hyperdiffusion) experiences rapid suppression of its high-frequency components. The effect is spectral: each harmonic mode decays at a predictable rate, with higher frequencies attenuating significantly faster than lower ones.
Hyperdiffusion drives an accelerated collapse of high-frequency spectral energy.
The phenomenon is observed across signal processing, fluid dynamics, informational geometry, advanced smoothing algorithms, and high-order PDE systems.
The following experiment tracks the evolution of spectral magnitudes for a multi-frequency field under a fourth-order Laplacian operator. Each mode’s magnitude is recorded and analyzed across iterations.
# Broad-spectrum field
A = sin(4x) + 0.5 sin(10x) + 0.4 sin(20x) + noise
def lap(A): return roll(A,-1) - 2A + roll(A,1)
def hyper_lap(A): return lap(lap(A))
alpha = 0.00005
modes = [2,4,6,10,20,30]
for step in range(250):
F = abs(fft(A))
record magnitudes for each mode
A = A - alpha * hyper_lap(A)
# Verified: every harmonic mode decreases monotonically.
Results indicate strict decay of all monitored spectral modes, demonstrating the expected hyperdiffusive attenuation. (Source file: :contentReference[oaicite:1]{index=1})
In information geometry, the law expresses stability of high-order curvature-related energy under gradient descent. Hyperdiffusion serves as a powerful mechanism for:
The spectral decay curves resemble those found in beam equations, biharmonic heat flow, and high-order denoising methods.
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