Two arithmetic classes of degree-(2,1) Trans-stratum continued fractions: a Birkhoff–Trjitzinsky / Gauss-continued-fraction dichotomy

Note: v3.0 retitles the work to reflect the proved Class A / Class B dichotomy, which generalises and supersedes the v1/v2 single-ratio conjecture. Version 3.0 of the T2B preprint (Trans-Stratum dichotomy of degree-(2,1) integer PCFs, Class A vs Class B) introduces four updates: two new mathematical findings and two consistency corrections established by the SIARC P-01..P-10 cross-paper coherence sweep. (1) The Class B 16-family count is qualified as identified-but-not-yet-saturated at dps=300, K=250 within the Worpitzky boundary, with saturation explicitly listed as open. (2) The Bauer 1872 orbit at a₂/b₁² = −1/36 (limits L = ±2k/log 2) is documented as a known closed-form orbit in the Log stratum (limits in Q·log 2), not a new stratum. (3) Theorem 3 is supplemented with a Painlevé III(D₆) reduction note: the BT discriminant D = 2·b_top² assigns all 16 Class B families uniformly to Painlevé III(D₆). (4) The Pure-regime corollary title is corrected from Q(π) to Q·π⁻¹ to match its body. Version 3.0 is coherent with UMB v3.0: five cross-manuscript inconsistencies identified by the P-10 audit (degree profile (2,2)→(2,1), tier-label collision, conjecture single-clause vs two-class form, Class B field universal vs Pure-regime scoped, and Cor. classB-pi title) are resolved across the UMB+T2B pair. v3.2 erratum (2026-06-08): Remark 13 restated — the BT-spectral locus (a₂/b₁²=+1/4, surd 3+2√2) is a locus fact, not a Sakai/Painlevé surface determination; surface type is OPEN (op:classB-surface-determination). Theorem 12 Wallis/4π reduction unchanged (proved).