Cyclotomic Arithmetic Dynamics of Catalan's Constant: Gauss's Period Calculus, Prime Transitions, and a Quadratic-Cone Representation

This preprint develops a cyclotomic-arithmetic reading of Catalan's constant. It realizes the alternating signs as a second-order readout of a four-step unit-circle orbit, introduces symmetric and antisymmetric fourth-order period states, and derives their finite sum-and-difference multiplication calculus. Multiplicative motion by odd primes is then separated into period-preserving and period-reversing transitions, yielding the exact amplitude decomposition G = (pi^2/8) product over primes p congruent to 3 modulo 4 of (p^2-1)/(p^2+1). The product identity is acknowledged as a classical consequence of Euler products; the contribution claimed here is its reconstruction as the global amplitude of an explicitly defined cyclotomic arithmetic motion. The paper also gives a divisor-orbit formula, a quadratic-cone representation for G^2, a four-sector rotation law, and a fixed-axis decomposition. The archive contains the compiled PDF, LaTeX source, release notes, metadata, manifest, and license text.