Finite-Time Collapse in Signed Sum-GCD Dynamics on Divisor Lattices

We introduce a signed arithmetic dynamical system on the divisor lattice of a positive integer n. For a divisor d of n and a sign epsilon in {+1, -1}, the map sends d to gcd(n, |d + epsilon n/d|). We prove that every orbit reaches a fixed point and construct a strict integer-valued Lyapunov function that gives an explicit stopping-time bound. The bound is sharp in arbitrarily large dimension. We completely classify the fixed points: d is fixed if and only if n = d^2 b and gcd(d, |1 + epsilon b|) = 1. We also derive pointwise bounds and exact prime-power formulas for the number of fixed points, construct arbitrarily long extremal transients using the Chinese remainder theorem, and prove that the mean number of fixed points satisfies sum_{n <= x} F_epsilon(n) = (zeta(2)/zeta(3)) x + O(sqrt(x) log x). The deposit includes the preprint, LaTeX source, bibliography, and Python scripts that reproduce the exhaustive computations and numerical tables using exact integer arithmetic.