The prime frontier of the Yellowstone permutation is linear

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), and for a prime p let n_p be the first index with p | a(n_p). We show that R(t) <= (2.715 + o(1)) t for every large t, that liminf n_p/p >= 0.3683, and, with a published result of Part II, that a(n) <= maxj<=n a(j) <= (2.715 + o(1)) n ln n. The argument has two steps. First (constant 5): at the entry step of a large prime P, the closure lemma forces many multiples of a small anchor prime to be already used, and those far above the current time must be among the few big terms; the rate for the number of big terms at level 10 proved in Part III beats the loss factor m/phi(m) of the mask. Second (constant 2.715): the level of that rate is lowered from 10 to 5.43 by a sharper Rankin-type count of masks with large m/phi(m), split by parity, and a low chain with a smaller shift. The bound is weaker than OEIS Conjecture 7 (a(n) <= (pi/2) n ln n): the constant is 2.715, not pi/2. All thresholds are ineffective and the result is purely asymptotic. We also record two reductions: a rate G_K(t) = o(t/lnln t) at one level K < pi would give Conjecture 7 for large n; and, conditionally on the OPEN hypothesis that the number of steps with two consecutive odd terms is at most 0.12 t asymptotically (Theorem PAR, conditional), R(t) <= (1.53 + o(1)) t and a(n) <= (1.53 + o(1)) n ln n. The growth gap (ii) (liminf n_p/p > 2/pi) and Conjecture 7 with the constant pi/2 are NOT proved unconditionally. A lower bound R(t) >= (2.08*10^(-3) - o(1)) t from an unpublished, once-refereed note is mentioned in a remark only; it is not proved here and not claimed. 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 drafted by an AI writer agent. Every proved statement was checked by two independent AI referee agents; 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) and Part III (doi:10.5281/zenodo.23557280, published together with this manuscript), which proves the few-big-terms rate and the fixed-time tools used here (including, in its appendices, the one-level inequality and label lemma they rest on). The parity count is proved in Appendix A. Side results of unpublished, once-refereed notes appear only in remarks and are not claimed.