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

The least prime κ(n) modulo which 0 has exact period n under the map x^2 + 1 is currently bounded below only by n log(logn), a consequence of Heath-Brown's recurrence theorem for quadratic iteration over finite fields. This work develops a method aiming at the stronger bound κ(n) ≫ n log n, conditional on two statements isolated below. The obstruction in Ren's image-size estimate for the family AX^d + C is identified precisely. Her main term has the right shape, of order p divided by the depth, but the exact empty-fiber detector has degree d^N in the depth N, and therefore forces moments of exponentially increasing order. The resulting geometric error is doubly exponential, which confines the effective depth to order log log p. The present method removes the detector rather than improving it. Only realized fibers are counted, each weighted by the reciprocal of its multiplicity, so that absence appears as the limiting endpoint of a graded weighting instead of being recognized by a discontinuous test. This yields an exact identity expressing image size as the expected reciprocal of one plus the accumulated lateral branching along a distinguished trajectory, and only the first four moments are then needed. Under exact critical period n, the branching loci of distinct depths are shown to be disjoint and the associated coverings linearly disjoint, giving a geometric orthogonality between depths. Multitemporal moments of order at most four then cost only exponentially in the total weighted depth, instead of doubly exponentially. An explicit quartic majorant and a geometric multiscale decomposition give a small-mass exponent strictly above 1, hence an image-size bound of order p divided by the depth, uniformly up to depths proportional to log p. Since every point of a critical cycle lies in every iterated image, κ(n) ≫ n log n follows, a gain of a full factor log n over log log n. The two remaining conditional statements concern the linear disjointness of distinct depths and the polynomial multitemporal counting; both have complete proof structures and require formalization in the language of function fields and normalizations. No published result implying the stated bound was found in the prior-art search of 25 August 2026.