Iterated Negative Complex Roots Principal-Branch Dynamics, Symbolic Itineraries, Exact Inverse Trees, Geometric Limits, Weighted Branch Measures, and Infinite-Depth Structure

We develop an exact and parameterized theory of iterated complex roots under explicit branch conventions. The principal negative-root recursion z n+1 = F r (z n) := Root pr r (−z n), r ≥ 2, separates completely into radial and angular dynamics. The radius obeys the exact law |z n | = |z 0 | 1/r n , while the principal argument is governed by a two-branch contraction. This yields a closed formula for the absolute angular variable, a global nonzero attracting two-cycle, a complete classification of periodic points, and sharp convergence estimates. We then enlarge the model to the phase-rotated family F r,β (z) = Root pr r (e iβ z) and prove an exact periodic-itinerary principle: every admissible finite branch word determines at most one periodic orbit, with explicit angle formula and multiplier. This broader family exhibits genuine higher-period behavior, including explicit period-three examples for r = 2, showing that the negative-root case is a distinguished symmetric member rather than a generic periodic regime. On the multivalued side we identify recursive root generation with inverse iteration of the monomial −z r , obtain exact regular-polygon formulas, exact Hausdorff distance to the unit circle, and equidistribution of normalized branch measures. We further introduce weighted branch measures, derive their limiting Bernoulli measures on the circle, and connect their entropy to information growth in the branch tree. Variable root orders are treated exactly at the algebraic level and quantitatively at the principal-dynamical level. Finally, we formalize the distinction among fixed-point, pointwise, set-valued, and measure-valued meanings of an infinite radical. The presentation is self-contained, but deliberately places the results in the classical literature on complex iterated radicals and the broader framework of complex dynamics and fractal measures.