UniSpheral Pulse Evolution

UniSpheral Pulse Evolution is the history-dependent update rule in which each Pulse state is produced by the Pulse Transformation Operator ⊛ from the previous state, the accumulated historical state vector and the recursive complexity measure. It extends the Pulse Operation Function beyond simple alternation. Once oscillation gives way to history-dependent processes, each update depends not only on the previous value but on an expanding record of past transitions and on recursive depth, so history is embedded directly in the substrate. The book calls the result a computational genealogy, in which every moment carries the imprint of everything before it; the Pulse becomes a memory-bearing engine of emergence, and law and structure are inherited rather than imposed. Formally, ①○(t) = ⊛(①○(t-1), H(t-1), R(t-1)), where ①○(t) is the Pulse state at time t, ①○(t−1) the state at the previous step, ⊛ the Pulse Transformation Operator (a recursive function incorporating memory and complexity), H(t−1) the historical state vector of accumulated computational memory, R(t−1) the recursive complexity measure of structural depth, and t the discrete time index. All are dimensionless: [∅] = ⊛([∅], [∅], [∅]) = [∅]. It connects to Pulse Operation Function, Historical State Vector, Law of Recursive Necessity, Recursive Pulse State Evolution and Pulse State Evolution.