Algebraic Morphogenesis Through Cochain Operators

Cellular automata and graph reaction–diffusion systems encode local spatial interactions in different mathematical forms. We develop a cochain-operator calculus for these two settings. Over a finite field F _q , every local rule on a finite neighborhood has a unique reduced polynomial representative. On an oriented line, the coboundary and endpoint maps recover the left and right shifts. Our main theorem shows that these operators, together with linear operations, constant cochains, and the degree-zero cup product, generate every finite-radius polynomial cellular automaton. Explicit formulas for Rules 30, 110, and 22 show how reflection-invariant linear coupling, directed transport, and nonlinear neighbor interactions enter the calculus. On a general graph, d ^* d is the unweighted combinatorial Laplacian and enters a graph reaction– diffusion recurrence. Over ℝ, the term − Dd ^* d with D ≥ 0 admits the usual diffusion interpretation; over F _q , the corresponding expression defines modular coupling without an intrinsic order. In the morphogenetic examples, we therefore distinguish pattern-generating dynamics from finite-state observation and use the Betti numbers of active induced subcomplexes to summarize observed patterns. This yields a common algebraic representation without identifying real-valued diffusion with finite-field dynamics.