gaussian-zeta

• Counts the fill of the diagonal design, corners (0,0) and (1,1), cell by cell on the side x side grid and checks the fluctuation rho(side) - 1/2 in exact rationals: 0 at even side, 1/(2 side^2) at odd side, -1/(2 side^2) for the orbit mate.
• Reduces Z(s) = (1/2) lambda(s+2) exactly to 1/192 and 1/1920 at s = 2, 4, then matches Z(2) = pi^4/192 and Z(4) = pi^6/1920 at 90 digits with pi by Machin against the library pi and lambda by Euler-Maclaurin over the odd integers against (1 - 2^-s) zeta(s).
• Sums the series straight off the counted fills, no closed form, and prints the gap at side <= 49, 199, 999.
• Checks the three parity shell identities on Z^2 from shell counts, evaluates Z_c(2) / (pi^2 G) for all fifteen nonempty base-2 designs, and checks each against a truncated lattice sum with the pi/(4R^2) tail.

RUN

uv run python research/lab/py/gaussian-zeta/gaussian_zeta.py

About 3 s. Domain is the source domain: fluctuation exact to side 80, series to side 999, shells to norm 1000000, lattice radius 6000.

WITNESSES

• pi.md:127-128 Z(2) = pi^4/192 = 0.50733901580... and Z(4) = pi^6/1920 = 0.50072353832..., both matched at 90 digits
• pi.md:132-136 exact to side 80, reductions 1/192 and 1/1920, two pi routines agreeing to the last digit, two lambda routes at 1.3e-82 and 5.2e-87, gap 8.3e-11 at side <= 999
• pi.md:139-142 orbit mate code 6, fluctuation -1/(2 side^2) on odd side <= 40
• bases.md:108-114 the three shell identities and the truncated lattice sums for all fifteen nonempty designs, a second pass over a wider domain than shell-energies, which is where the page number on those lines comes from
• bases.md:120 Z_c(2) / (pi^2 G) = (a_ee + 3 a_oo + 6 a_eo + 6 a_oe) / 24
• bases.md:124 fifteen rationals 1/24 to 2/3
• bases.md:126 13 pi^2 G / 24 = 4.8967847822

NOTE

• The 9.4e-10 at bases.md:114 is the shell-energies truncation at norm 200000; at radius 6000 with the tail added the worst gap here is 1.2e-13, so the two agree on the identity and differ only in domain.
• The 4D grading at bases.md:137-144 is not computed here.