UniSpheral Recursive Growth Relations

The UniSpheral Recursive Growth Relations are the increment and amplification relations of the Recursive Growth Law: each completed pulse adds as much data as all earlier pulses together, and the ratio between successive levels is fixed at two. They state the doubling of ℜ(n) = 2ⁿ explicitly. The Binary Increment Law, ℜ(n+1) − ℜ(n) = ℜ(n), says pulse n + 1 adds exactly what every earlier pulse holds. The Amplification Factor, A(n) = ℜ(n)/ℜ(n−1) = 2, says the amplification never tapers: it is two at every depth from n = 1. The first count is not an amplification but a genesis: the Zinf's one whole completion, ℜ(0) = 1, from which every doubling follows, so the Increment Law holds from n = 0 (2 − 1 = 1). Capacity doubles at every step with nothing added that the binary pulse does not supply, so growth is bounded not by a slowing of the doubling but by the substrate. The older square relations, rate 2(n + 1) and constant second difference 2, belong to quadratic structural capacity in Quantum Pulse Theory. Formally, ℜ(n+1) − ℜ(n) = ℜ(n) and A(n) = ℜ(n)/ℜ(n−1) = 2 (n ≥ 1), where ℜ(n+1) − ℜ(n) is the data added by pulse n + 1, A(n) the doubling ratio, ℜ(n) and ℜ(n−1) the total data capacity after n and n − 1 completed pulses, and n the count of completed pulses; all [∅]. It connects to Recursive Growth Law, Amplification Factor, Amplification Factor Properties, Substrate Capacity Limit Properties and BPT Foundational Equation.