The Yellowstone permutation, III: few large terms

This manuscript is Part III of a series on the Yellowstone permutation a (OEIS A098550). Let G_K(t) be the number of n <= t with a(n) > K t. Part I proved a(n) <= n (ln n)^7, and an argument of the same program, reproduced here in Appendices A and B, gives limsup G_K(t)/t <= 74 K^(-1/2) + 2.1*10^6 K^(-1) for every fixed K >= 1156. We show that the large terms are few already at the level 10t: G_14(t) <= 153 t (lnlnln t)/(lnln t)^2 and G_10(t) <= 6*10^7 t (lnlnln t)/(lnln t)^2 for all large t, with an ineffective threshold. In particular {n : a(n) > K n} has density 0 for every K > 14. The tools are a Cauchy-Schwarz bound for the values that the closure lemma of Part I forces into a window, a sieve count of the anchors in a band, and a bootstrap in the level K. We also show that the sum of a(n)/phi(a(n)) over n <= t with a(n) > 10t is o(t), and deduce a bound (5 lambda_Y + o(1)) t for the total weight v/phi(v) of the used odd values v with v/phi(v) >= Y. Two reductions of the linear prime frontier (and hence of OEIS Conjecture 7) are stated as implications only, without proof. Conjecture 7 is not proved here. AI-assistance and review status: The results were developed with an AI system (Anthropic Claude). Each result taken from the source working papers was checked by two independent AI referee agents (separate Claude instances that had not written the text under review); details are in Section 9 of the manuscript. Appendices A and B were assembled for this version from refereed source texts (with the referees' fixes applied) and were then refereed as a whole in this form by one further AI referee agent. No human peer review has taken place. All thresholds are ineffective. Dependencies: The proofs use Part I (doi:10.5281/zenodo.23437783) and Part II (doi:10.5281/zenodo.23449375) and classical results (Rosser-Schoenfeld, Robin, Mertens, Legendre, Selberg's sieve, the fundamental lemma of sieve theory). The few-large-terms input of an earlier unpublished working paper (TOPYS) and the label lemma it uses (SOFTMASK) are proved in Appendices A and B; the unpublished FHYS Lemma P that TOPYS cited is replaced by Parts I and II. The two implications of Section 7 depend on further unpublished notes and are not claimed.