On the touching arc of the snub cube: a closed form and an independent check

This note gives the closed form theta_* = arctan(2 - t), where t is the tribonacci constant, for the endpoint of the initial touching arc of the snub cube established in the preprint by L, W. An independent certificate combines exact arithmetic in Q(t) with outward-rounded Arb interval arithmetic to verify closed shadow containment along the arc and its failure immediately beyond the endpoint within the stated parameter range. An explicit coordinate change identifies this family with the tie described in Tom 7's noperts repository and distinguishes its angular and Cayley parameters. The resulting endpoint shows that the approximate range 0.165 in the inspected repository descriptions exceeds the initial touching range under either parameter interpretation. Two additional remarks address a weight-sign inference in Lemma 5.2 and the interpretation of a binary64 bound in Appendix A.4 of the L, W preprint. The scope is this local family and these textual points; the note does not reproduce a global non-Rupert certificate.