Static Power Dynamics and Harmony Cycles: Universal Convergence and Attractor Classification in Iterated Digit-Power Mappings

Static Power Dynamics and Harmony Cycles:
Universal Convergence and Attractor Classification in Iterated Digit-Power
Mappings

We introduce and investigate Static Power Dynamics, a novel class of iterated digit-power mappings
defined by the transformation
fP(n) = Σ di^P,
where di are the decimal digits of n ∈ ℕ and P ≥ 2 is a fixed integer parameter applied uniformly and
statically to all digits. Utilizing a dedicated iterative computational engine (the Sufi Cycle Finder [10]), we
systematically classify all periodic attractors — termed Harmony Cycles — for parameters P ∈ {2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 15, 100} across ranges up to 100,000. The mapping recovers the classical Happy
Numbers system as the special case P = 2, and its non-trivial fixed points at each P are precisely the
generalized narcissistic (Armstrong) numbers (Theorem 1). We prove analytically that n = 1 is a universal
fixed point for all P ≥ 2, establish a boundedness theorem guaranteeing eventual periodicity, and document
a striking even/odd parity pattern in attractor counts. A complete attractor landscape is provided for each
tested parameter, revealing rich non-monotone structural complexity as P varies.