A continuous-state cellular automaton on [0,1]³ where each cell carries an RGB vector. One axis — Blue — is subject to preferential decay through habituation, while a competing error-detection mechanism restores Blue when local chromatic discrepancies are large. The balance between these forces produces a sharp phase transition: below a critical decay rate, Blue persists and chromatic diversity is maintained. Above it, Blue collapses and the entire population converges to the Yellow vertex (1,1,0) — maximum capability on two axes, zero on the third.
The post grounds the dynamics from the differentiation operator D(v) on {0,1}³ (Posts 1-2), demonstrates the emergent phase transition through bifurcation analysis, estimates the critical exponent, and visualizes the population trajectory through the binary cube as it falls from random initialization toward a specific vertex attractor. The terminal state histogram shows Red and Green distributed broadly across moderate-to-high values while Blue is compressed into a narrow spike near zero — the chromatic signature of a system that has lost its inhibitory axis.
Five questions for the community:
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Inevitability of collapse. In any three-axis system where one axis is subject to preferential degradation while the other two are not, is convergence to the two-axis vertex inevitable above some critical decay rate? Or can system topology — lattice geometry, boundary conditions, coupling structure — prevent the collapse entirely?
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Universality class. The critical exponent β characterizes the universality class of the phase transition. Is the transition in the same universality class as known lattice models (Ising, percolation), or does the three-axis chromatic structure produce a novel class?
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Negative feedback existence. The feedback loop (Blue decay reduces diversity → reduced diversity weakens error signals → weaker signals accelerate Blue decay) is self-reinforcing above the critical threshold. Is there a corresponding negative feedback loop below the threshold that actively stabilizes Blue, or does the subcritical regime simply lack positive feedback without possessing negative feedback?
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Yellow maximization. The terminal attractor is the Yellow vertex (1,1,0), not an arbitrary point on the B=0 face. The surviving components are driven to their maximum values. Is this a general property of inhibitory decay in coupled systems — that the loss of the inhibitory axis maximizes the uninhibited axes — or is it specific to the coupling structure defined here?
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Initial condition dependence. The system was initialized with uniform random RGB values. Does the critical threshold δ_c depend on initial conditions? Specifically, if the population starts near the Blue vertex (0,0,1), does δ_c increase, suggesting that initial proximity to the inhibitory axis provides transient protection against its decay?
Notebook included.
https://www.wolframcloud.com/obj/b1180013-2f9d-4af8-a808-8462ddceeb8d
Inhibitory Decay.nb (3.01 MB)