Generative Structural Theory v1.9: Finite Simulation Rank, Tightness and Certificate Inheritance

GST v1.9 does not prove that d(n) is unbounded. That sentence is the paper's own, and the rest of this description says what is proved instead and what the question has been reduced to. We study the rank of a finite simulation game on cut chains: die(X,Y) is the number of approximant rounds a pair survives, and d(n) its maximum at ambient size n. PROVED, for arbitrary fixed n: a recursion for die and its forall-exists form; two size bounds, die(X,Y) <= |X| - 2 and the dual die(X,Y) <= |Y| - 1, whence d(n) <= n(n-1) - 2; a reduction showing that finite rank is decided entirely on the boundary of the maximal simulation M, together with what tightness forces there; that every fan carries a single value of the N4 flag, with the consequence that whether a fan contains an M-reply can be decided on its inclusion-maximal members; a diagonal lifting rule with its exact reach; and an embedding lemma — an isolated enlargement of the ambient size preserves die exactly, so that enlarging the universe is free and only the certificate inside it is at issue. COMPUTED: d(4) = 7 exactly, and certified lower bounds d(5) >= 10 and d(6) >= 10. REDUCED, not proved: the unboundedness question is reduced to a single statement — the existence of one cross-size family together with one partial global positional selection, consistent with an explicitly described policy, that is self-consistent with the least obligation fixpoint it generates. Such a selection is exhibited up to rank 10: 139 states, 303 entries, one state answering 849 distinct left cuts. Thirteen routes that provably cannot close the question are recorded at three distinct strengths — class barriers, exact refutations, and routes whose tried formulations failed — and the three are kept apart. CONVENTIONS, visible throughout the paper and the deposit: every count is published with the search space that produced it, and no trend is read from two or three values; a run stopped by a budget is recorded as stopped and contributes nothing, never as a negative result; controls declare in advance whether they are expected to pass or to fail; where a reading was withdrawn, the withdrawal is in the text rather than in a footnote. The deposit archive contains the paper, its complete LaTeX source, the plain-text draft, the 37 working records with every script and output, the rank-10 witnesses, and a manifest and verifier. The verifier checks every file against the manifest and then deliberately breaks each check to show that it fires. v1.9 is a sequel to v1.8 (doi:10.5281/zenodo.22348090), not a correction of it.