Variable arithmetic periods : A Multiscale Generalization of Ren's Image-Size Method to Logarithmic Depth Using Harmonic Fiber Weighting

Abstract

This work develops a new method for estimating the size of iterated images of the quadratic map sending x to x squared plus one over finite prime fields, with the aim of controlling the smallest prime carrying a prescribed exact critical period. The approach departs from Ren’s method at a decisive point. Ren detects empty fibers exactly, which requires polynomial detectors of exponentially increasing degree and moments of exponentially increasing order. The resulting geometric error grows doubly exponentially with the iteration depth and restricts the effective range to depths of order the logarithm of the logarithm of the prime. Instead of detecting empty fibers, the present method counts only realized fibers and weights them by the reciprocal of their multiplicity. In this harmonic formulation, absence no longer has to be recognized by a discontinuous detector: the zero weight of an empty fiber appears naturally as the limiting endpoint of a graded weighting of nonempty fibers. This leads to an exact identity expressing image size through the harmonic mean of fiber multiplicities. The iteration depths are then divided into geometric blocks. Only the first four moments are required. Distinct depths are separated through their branching loci, and the corresponding coverings are shown to be linearly disjoint, providing a form of geometric orthogonality between depths. This permits multitemporal moment estimates whose geometric cost remains only exponential in the total depth. A quartic majorant and a multiscale small-mass argument then give sufficient contraction to reach depths proportional to the logarithm of the prime. The resulting candidate theorem gives an image-size bound of order the size of the field divided by the iteration depth throughout this logarithmic range. Applied to a critical orbit of prescribed exact period, it yields a candidate lower bound of order the period times its logarithm for the smallest prime supporting that period. The main innovation is therefore not an improvement of Ren’s exact fiber detector, but its replacement by a harmonic, multiscale and low-moment method that makes logarithmic depth accessible.