A Galerkin transfer-graph criterion for BKM occupation-tail integrability

This paper introduces a conditional criterion reducing the Beale–Kato–Majda regularity test for 3D Navier–Stokes Galerkin approximants to two verifiable conditions on a shell-to-shell transfer tensor: (1) a transfer-race margin threshold η* > 21/5−1, measuring viscous exit dominance over retained nonlinear feeding, and (2) a weighted-certificate exclusion of recurrent high-shell transfer cores via joint-spectral-radius subcriticality. The paper proves an unconditional occupation split (Proposition 2.1′), a narrow-band dissipative-range lemma at depth M ≥ 5 with explicit constant C★ = 1280 (Lemma 7.4.2), and the full θ-certificate closure under computable profile conditions (Prop. S2.15). It does not prove unconditional regularity; the remaining analytic targets — mid-depth interpolation and the anti-coherence bound — are named, localized to a finite shell belt, and tested in a GPU proof-lab (A100, grids 32–256, Re up to 3142). Appendices calibrate the formalism against known thresholds: the (d,s)-phase diagram recovers 2D regularity and the Lions critical hyperviscosity s = 5/4 as zero loci of a single function; self-similar blow-up profiles are shown to require nonlinear efficiency ≥ 6.725×; and a Duhamel L∞ ledger provides unconditional per-event costs for coherence-driven escalation. Code, data, and SHA-256-linked verification scripts are included.