The prime frontier of the Yellowstone permutation falls behind time: limsup R(t)/t < 1

This manuscript continues the series on the Yellowstone permutation a (OEIS A098550). Let R(t) be the least odd prime with no multiple among a(1), ..., a(t), for a prime p let n_p be the first index with p | a(n_p), and let E(t) be the number of even terms among a(1), ..., a(t). The previous manuscript of the series proved R(t) <= (2.715 + o(1)) t. We prove that the prime frontier eventually falls behind time: limsup R(t)/t <= 0.9265 < 1, liminf n_p/p >= 1/0.9265 > 1.0793, and liminf E(t)/t >= 0.2440; hence, with the frontier corollary of Part II, a(n) <= maxj<=n a(j) <= (0.9265 + o(1)) n ln n. In particular the inequality a(n) <= (pi/2) n ln n of OEIS Conjecture 7 holds for all n >= n_0, where n_0 is ineffective. These constants use a smooth-part Rankin count for the masks, for which two independent proofs are given (Appendices A and B). With Rankin counts only, the same method gives limsup R(t)/t <= 1.045 < pi/2, liminf n_p/p > 0.9569 > 2/pi and liminf E(t)/t >= 0.2301, so liminf n_p/p > 1 is not reached by Rankin counts alone. On the way the manuscript proves the intermediate bounds on limsup R(t)/t: 2.405, 4.625/3, 1.2275, 1.215 and 1.155 (Rankin counts) and 2.14, 4.025/3, 1.1075, 1.08, 2.96/3 and 0.9625 (smooth-part count). The new ingredients are a hole sieve (an even number 2k still unused at time t forces every earlier step with an even anchor and a larger value to have a mask sharing an odd prime with k, by the closure lemma of Part I), the remark that a used odd value above the odd-mask level either has an odd mask or follows an even term, a low chain with a Rankin count at one fixed exponent, few-big-terms rates split by the parity of the mask, an elementary parity count that yields the lower bound on E(t), a split of the forced band at the entry of a prime, and a split of the times into classes according to E(t)/t. All thresholds are ineffective and the results are purely asymptotic; the numerics are double precision with outward-rounded discretisation, not interval arithmetic. Conjecture 7 for all n is NOT proved, no explicit n_0 is given, and a proof of liminf n_p/p >= 1 by Rankin counts alone, as well as the parity bound liminf E(t)/t >= 0.275 used by a conditional result of the program, remain open. AI-assistance and review status: The mathematical content was produced by AI agents (Anthropic Claude) in the author's research program on OEIS A098550, and the text was written by an AI writer agent from the program's internal notes and referee reports. Each section taken from a program note follows a note checked by two independent AI referee agents (separate Claude instances that had not written the text under review), except Appendix A (one full referee report plus two independent re-derivations; the same count has a second, twice-refereed proof in Appendix B); a few lemmas assembled by the writer were checked only by the referees of the assembled text. Each successive assembled version was in addition refereed as a whole by two further AI referees (mathematics; citations, status claims and wording), with all fixes applied; details are in Section 14 of the manuscript. All referee reports cited are AI-generated. No part has been reviewed by a human mathematician. Dependencies: Part I (doi:10.5281/zenodo.23437783), Part II (doi:10.5281/zenodo.23449375), Part III (doi:10.5281/zenodo.23557280) and the previous manuscript on the linear prime frontier (doi:10.5281/zenodo.23558450), and classical results (Rosser-Schoenfeld, Robin, sieve methods as in Halberstam-Richert). One unpublished program result (GIIRG Lemma EV, refereed twice) is used only for the lower bound on R(t)/t and the upper bounds n_p <= (8.197 + o(1)) p and their analogues; the main bounds above do not use it. Conditional statements that rest on further unpublished notes appear only as remarks and are not claimed.