An affine Singer layer on F2^6, its (T,D,K) triple, and = AGL(1,64)

This preprint studies an affine Singer layer on F_2^6 and its associated triple (T,D,K), where D = (id xor T)^(-1) and K = (id xor D)^(-1). The map T has one fixed point and one 63-cycle. The triple has cycle types D = 1+63 and K = 1+7*9, with fixed points {38}, {41}, {63}, and satisfies (id xor K)^(-1) = T. We prove that the generated group is = AGL(1,64), of order 4032 = 64*63. The construction is also treated uniformly for 2k-bit words. The required even-characteristic primitive trace-one input is proved self-containedly in Appendix B using Gauss sums and a deterministic finite certificate for k <= 29. The underlying theorem is classical: it is the n=2, t=1 case of Cohen (1990, Theorem 1, p. 1), while Moreno (1989) proved the trace-one quadratic case. Cohen (2003, Theorem 1.5, p. 170) gives an equivalent square-order formulation. No new priority is claimed. The record contains the manuscript PDF, the LaTeX source, the deterministic verifier verify_k1_29.py for the finite certificate, and a detailed supplementary proof archive.