A chromatic cellular automaton on {0,1}³ generates a cruciform pattern from a single seed (established in Post 3). At every timestep, the null-space complement of the pattern — obtained by mapping each state s → 7−s — is computed and compared to the positive space using five independent complexity measures: Shannon entropy, joint entropy of adjacent pairs, spatial mutual information, 2×2 block entropy, and Kolmogorov complexity approximation via compression. Result: the first four measures show exact equality C(P,t) = C(N,t) at every step. The fifth (Kolmogorov complexity approximated via compression) shows near-equality with fractional difference under 3%, consistent with encoder-dependent variation in a computable upper bound rather than genuine structural asymmetry. The structural complexity of the complement matches the complexity of the pattern — not approximately, but identically — across all information-theoretic measures tested.

The complement operator also reveals a vertex absent from the automaton's lifecycle. The lifecycle visits Red, Green, Blue, Yellow, and Magenta. Cyan (0,1,1) never appears. In the complement, Red maps to Cyan. Since Red is the most frequently renewed state (Dead → Idea), Cyan becomes the dominant active state in the null space. The vertex absent from the dynamics is the most prominent vertex of the shadow.

Four questions for the community:

1. Continuous extension. The information symmetry C(P,t) = C(N,t) holds by construction for any bijective complement on a finite vertex set. Does it extend to continuous state spaces on [0,1]³?

2. Cyan dominance. The complement reveals Cyan (0,1,1) as the dominant active state of the null space. If Cyan were introduced into the lifecycle, would the complement become impoverished at a corresponding vertex?

3. Simultaneous threshold crossing. The null-space complement achieves identical structural complexity without its own dynamics. In any system where the positive space crosses a self-referential complexity threshold, does the complement cross simultaneously?

4. Universality challenge. Five complexity measures all show C(P,t) = C(N,t). Is there ANY computable complexity measure for which the equality fails?

Notebook Attached.

https://www.wolframcloud.com/obj/f17551bc-aec2-415b-9323-ad8865b5d46d

Attachments:

Information Symmetry.nb