eisenstein-stack

• The hexagonal twin of lab/py/spun-stack: the exact spun stack whose layers are the associate classes of the Eisenstein integers Z[omega], omega = e^(2 pi i/3), norm N(a + b omega) = a^2 - a b + b^2, six units, unique factorisation.
• Layer z is the lattice z^-1 Z[omega], scale |z| and rotation -arg z in one multiplication, one layer per nonzero associate class with the zero class excluded.
• Coordinates are taken in the basis 1, omega, so the fundamental domain of C/Z[omega] is the unit square of that chart and the picture is its image, the fundamental parallelogram. Every node is a pair of Fractions in that chart; the arithmetic is exact throughout, with nearest-integer Euclidean gcd in Z[omega] and the cyclotomic field Q(zeta_360) for the whole-degree rotations.
• The constant the base-3 page carries, zeta_K(2) = zeta(2) L(2, chi_-3), is the constant in this stack's node count.

WHAT IT PRINTS

• field_rotation_degrees reduces zeta^d + zeta^-d and (zeta^d - zeta^-d)(zeta^120 - zeta^240) modulo Phi_360 and returns the degrees where both remainders are constant, that is where cos is rational and sin is a rational multiple of sqrt 3: 0, 60, 120, 180, 240, 300, six of 360.
• spot_check_degrees re-decides twelve of those degrees through sympy.minimal_polynomial on cos(d deg) and sin(d deg)/sqrt 3 and agrees with the cyclotomic route on all twelve; 90 and 270 are the instructive failures, rational cosine and irrational sin/sqrt 3.
• rotation_hits checks that all 58 solutions of p^2 + 3 q^2 = r^2 with r at most 60, read as the pair (cos, sin/sqrt 3) = (p/r, q/r), are of the form w/conj(w) = w^2/N(w) for an Eisenstein w in the box of side 20.
• hex_classes_closed against hex_classes_direct for t = 0..400: equal throughout; the first twelve values of the hexagonal circle count are 1, 1, 2, 3, 3, 3, 5, 5, 6, 6, 6, 7.
• literal_stack builds the stack at norm bound 50 by exact stacking of all 31 layers and finds 630 nodes, equal to the Eisenstein totient sum sum Phi(d) over the same 31 classes from totient_sum.
• closed_brightness reads each node from its reduced Eisenstein denominator alone, computed by reduced_denominator, as h(floor(N/N(d))) with h(t) = sum_j (floor(t/(3j+1)) - floor(t/(3j+2))): 630 comparisons against literal stacking, 0 mismatches, the origin at 31, which is also the maximum.
• totient_sum gives 630, 9606, 151020, 337026, 945486 and 2419950 nodes at norm bounds 50, 200, 800, 1200, 2000 and 3200, ratios to N^2 of 0.252000, 0.240150, 0.235969, 0.234046, 0.236372 and 0.236323; main prints each deviation from the limit scaled by N/log N, +0.214, +0.186, +0.090, -0.198, +0.304, +0.438, bounded and of both signs.
• main prints the constant's factors at 30 working digits through the Hurwitz zeta form L(2, chi_-3) = (zeta(2, 1/3) - zeta(2, 2/3))/9: zeta(2) = 1.644934066848226, L(2, chi_-3) = 0.781302412896486, zeta_K(2) = 1.285190955484149, the residue pi/(3 sqrt 3) = 0.604599788078073, and the limit pi/(6 sqrt 3 zeta(2) L(2, chi_-3)) = 0.235217881630015.
• csl_zeta_ratio expands (1 + 3^-s)^-1 zeta_K(s)/zeta(2s) by Dirichlet division to bound 100 and csl_euler_product expands prod_{p = 1 (3)} (1 + p^-s)/(1 - p^-s) over the same bound: equal, with nonzero coefficients 1 at 1, 2 at 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 97 and 4 at 91.
• draw stacks the 774 lit nodes at norm bound 60, brightest 35 at the origin, into eisenstein-stack.png, dots sized by the square root of brightness inside the fundamental parallelogram.

RUN

• uv run python research/lab/py/eisenstein-stack/eisenstein_stack.py
• From the repo root. One core, about three seconds.
• Domain is the full source domain: all 360 whole degrees, the circle count on t = 0..400, the stack at norm bound 50 node by node, totient sums to norm bound 3200, the coincidence series to bound 100.
• Writes eisenstein-stack.png beside itself and nothing else.

WITNESSES

• The Eisenstein address: literal_stack against closed_brightness and totient_sum, 630 nodes and 0 brightness mismatches at norm bound 50.
• The hexagonal twin of floor(N/b): hex_classes_closed against hex_classes_direct on t = 0..400.
• The hexagonal rational rotations: field_rotation_degrees and spot_check_degrees give the six whole degrees, rotation_hits the 58 rational rotations as w/conj(w).
• The constant: totient_sum at six norm bounds against the printed pi/(6 sqrt 3 zeta(2) L(2, chi_-3)), with the scaled deviation.
• The coincidence census: csl_zeta_ratio against csl_euler_product.