Shell Energies

• The three exact shell relations behind the base-2 design energy Z_c(s), as integer identities on the Dirichlet coefficients of r2.
• Every parity class of Z^2 counted by direct lattice enumeration to norm n = 200000, r2 checked against 4 (d1 - d3).
• Doubling gives r2_ee(n) = r2(n/4); the 45 degree rotation gives r2_oo(n) = r2_mix(n/2); the coordinate swap gives r2_eo = r2_oe.
• The rotation also run as a map: every odd-odd point of every even norm to n = 6000 sent to ((i+j)/2, (i-j)/2), the image required to be mixed parity of half the norm and the map required bijective onto that class.
• The four linear relations solved symbolically to recover Q_c(t) = a_ee t^2 + a_oo t (1-t) + (a_eo + a_oe)(1-t)/2.
• The closed form Z_c(s) = 4 zeta(s) beta(s) Q_c(2^-s) against truncated lattice sums at s = 2, 3, 4 for all fifteen nonempty designs, the s = 2 comparison adding the leading tail k pi / (4N).
• beta(s) taken from the Hurwitz zeta and self-tested against Catalan and pi^3/32.

RUN

• uv run python research/lab/py/shell-energies/shell.py
• Under two seconds on one core; prints only, writes nothing.

WITNESSES

• bases.md:109 S_ee = t^2 * S as r2_ee(n) = r2(n/4), 0 mismatches to n = 200000.
• bases.md:110-111 S_oo = t * S_mix as r2_oo(n) = r2_mix(n/2), 0 mismatches to n = 200000, with the rotation bijective on 4716 odd-odd points to n = 6000 and 0 faults.
• bases.md:112 S_eo = S_oe as r2_eo(n) = r2_oe(n), 0 mismatches to n = 200000.
• bases.md:113-115 the same n = 200000 domain, and the worst gap 9.4e-10 at s = 2; beyond the page, 3.9e-11 at s = 3 and 1.3e-16 at s = 4.