# 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.