The Eisenstein stack

• 2026-09-09 [Proved] The Eisenstein spun stack is the hexagonal twin of the Gaussian one: layers are the nonzero associate classes of Z[omega], layer z the lattice z^-1 Z[omega]; the hexagonal circle count is the floor sum h(t) = sum_j (floor(t/(3j+1)) - floor(t/(3j+2))), the twin of the Gaussian floor(t/(4j+1)) - floor(t/(4j+3)), equal to direct enumeration of classes for every t from 0 to 400; a node of reduced denominator class [d] has brightness h(floor(N/N(d))), checked by literal stacking of all 31 layers at norm bound 50 in exact rational coordinates, 630 nodes and 0 mismatches, the origin at 31; the lit set has sum_{[d], N(d) <= N} Phi(d) points with Phi(d) = N(d) prod_{p | d} (1 - 1/N(p)), reading 630, 9606, 151020, 337026, 945486 and 2419950 at norm bounds 50, 200, 800, 1200, 2000 and 3200. Witness: lab/py/eisenstein-stack hex_classes_closed, hex_classes_direct, literal_stack, closed_brightness, totient_sum.
• 2026-09-09 [Proved] A rotation keeps two hexagonal layers coincident iff it lies in Q(sqrt -3), iff its cosine is rational and its sine a rational multiple of sqrt 3, and those rotations are exactly w/conj(w) for nonzero w in Z[omega] by Hilbert 90 (all 58 rational rotations of denominator at most 60 are reached from the box of side 20); at whole degrees exactly the six multiples of 60 survive, against four of 360 on the square lattice, by cyclotomic reduction modulo Phi_360 and twelve minimal-polynomial spot checks; the hexagonal coincidence series is (1 + 3^-s)^-1 zeta_K(s)/zeta(2s) = prod_{p = 1 mod 3} (1 + p^-s)/(1 - p^-s), read at source in Pleasants, Baake and Roth and re-derived coefficient by coefficient to bound 100. Witness: lab/py/eisenstein-stack rotation_hits, field_rotation_degrees, spot_check_degrees, csl_zeta_ratio, csl_euler_product, REFS.md.