A fixed birational map without a prime-field cycle-distribution limit

For the fixed birational map L(u,v)=(u,(u²+1)v), every prime p congruent to 3 modulo 4 gives a permutation of the full affine plane F_p². With point-weighted periods normalized by p, the distribution function at every fixed 1/2≤x<1 equals 1−φ(p−1)/(p−1)+o(1), has limsup 1 and liminf at most 23/24. Neither these CDF values nor the period probability measures converge on any further admissible prime subset of positive upper relative density. A separate proposition covers every odd prime on the invariant regular domain of size p(p-r), where r is zero or two. The primitive principal term is (p-r)φ(p−1)/(p−1). Under uniform-point deletion or arbitrary finite nonnegative assigned periods only on exceptional fibers, bounded statistics change by O(1/p). The regular CDF and a globally bounded continuous ramp thus exclude convergence on every positive-upper-prime-density selection under these conventions. Completed measures are on nonnegative reals; assigned periods may exceed p. Primes congruent to 1 modulo 4 still do not give full-affine permutations. This is a precisely scoped application to the broad existence assertion in Vivaldi's 2004 Oberwolfach contribution, joint work with Roberts. The triangular family, quadratic primitive-value machinery, shifted-prime totient mean and continuous scalar empirical law are established and credited to the actual sources. The positive-density selection conclusion uses the old atomless scalar law, not merely the existence of two subsequences. No new map family, character-sum theorem, arithmetic mean, scalar law, universal elliptic theorem, polynomial-automorphism theorem or prime-average cycle theorem is claimed. The bounded priority audit authenticated no earlier statement of this exact application in the inspected sources. The cited Neumärker–Roberts–Viallet–Vivaldi 2011 preprint body and version relationships remain unrecovered; no inference about its unseen contents is made. No worldwide novelty, current-open status or historical firstness is claimed. The source/verification ZIP preserves all 17 original research files and historical mathematical audit artifacts, supplies exact manifests and a new dependency-free standard-library checker. The latter passed 147,619 finite exact checks over 87 admissible primes through 1000, eight full affine orbit graphs with 6,088 points, six bad-reduction controls and a totient bound through 10,000. An additional separate checker passes 46,298 exact controls over all 167 odd primes through 1000 (80 class1,87 class3), including 10,178 raw regular graph points and exceptional assignments beyond p. Historical 8,347,742 finite assertions and 230,780 analytic controls are separate retained runs. Original SymPy scripts were not successfully replayed in this preparation; the analytic proof and explicitly cited classical inputs establish the infinite-prime statements. AI systems were extensively involved in research, model and proof development, computation, literature investigation, writing and adversarial verification. This is an unrefereed preprint with no formal independent human peer review. Internal AI audits do not constitute human peer review.