SANDPILE

Self-Organized Criticality: why hoarding always collapses

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4+ avalanche!
Grains fall at random positions

Statistics

Grains dropped0
Avalanches0
Largest avalanche0
Average height0.00
Max cell0
Std deviation0.00
Grains lost (edge)0
The threshold is 4. No cell can hold 4 or more grains — it topples, sending one grain to each neighbor. At edges, grains fall off the table.

The system self-organizes to a critical state where the average height converges to ~2.1. Small avalanches are common, large ones are rare — a power law. Hoarding is structurally impossible.

Avalanche Size Distribution

The axiom connection: threshold = 4 = K+sigma = spacetime dimensions. Each topple redistributes to 4 neighbors (up/down/left/right = D*D directions). The grid = Z/20 x Z/20. Boundary loss = OMEGA projection (grains that leave don't return). Power law = spectral gap selecting for equality.

Contrast Panel

Standard View
Axiom View
SOC is an empirical phenomenon
SOC is spectral gap enforcing equality
Power laws "just happen"
Power laws = eigenvalue distribution of ring Laplacian
Threshold 4 is arbitrary
4 = K+sigma = minimum for spacetime closure
Redistribution is a model choice
Redistribution IS the D-bridge: distinction shares to neighbors
Cooperation is a nice social idea
Cooperation is mathematically enforced by spectral gap > 0
Avalanches are destructive events
Avalanches are sigma restoring balance: sigma*n = n