eisenstein-visibility

• Recomputes the base-3 page constants at 60-digit working precision, printed to 43 digits: pi by Machin, trigamma psi1(x) = sum 1/(n+x)^2 by Euler-Maclaurin in exact rationals (100 explicit terms, Bernoulli tail through B_20), then L(2, chi_-3) = (psi1(1/3) - psi1(2/3))/9, Catalan G = (psi1(1/4) - psi1(3/4))/16, zeta(2) = pi^2/6, zeta_K(2) = zeta(2) L, zeta_Qi(2) = zeta(2) G and both reciprocals.
• Cross-checks by the reflection identity psi1(1/3) + psi1(2/3) = 4 pi^2/3, prints the implied Cl2(pi/3), and runs the two-term relation search as continued fractions of L sqrt(3)/pi^2 and L/pi^2 with G/pi^2 and zeta(2)/pi^2 as controls, reporting each last convergent denominator.
• Repeats the same constants by independent float routes: the paired Dirichlet series for the L-function, the Eisenstein lattice sum (1/6) sum 1/N(z)^2 and the Gaussian (1/4) sum 1/(a^2+b^2)^2 truncated at norm 4e6 with their analytic tails, and the gcd(a,b) = 1 census of [1,3000]^2.
• Builds Z[omega] from omega^2 = -1 - omega (norm a^2 - a b + b^2, nearest-integer division by the conjugate, Euclidean gcd), sieves 400000 seeded random pairs for the coprime fraction, checks base^level - base^(level-1) against phi(base^level) for base 2, 3, 5 and the base 6 counterexample, and draws the hexagonal visibility figure by per-pixel coverage blending.

RUN

uv run python research/lab/py/eisenstein-visibility/eisenstein_visibility.py then uv run python research/lab/py/eisenstein-visibility/makefig.py

About 3 s and 0.1 s. Domain is the source domain: prec 60, 200000 Dirichlet pairs, lattice norm <= 4000000 at radius 4400, census [1,3000]^2, sieve 400000 pairs in [-500,500]^2, figure 880x760 over [-30,30]^2.

WITNESSES

• bases.md:16 6/pi^2 = 0.607927...; bases.md:17 the 3000 x 3000 grid gives 0.608042
• bases.md:51 forty digits asked, 43 printed: bases.md:59 1.644934066848226, bases.md:60 0.781302412896486, bases.md:61 1.285190955484149, bases.md:62 0.778094489175179, bases.md:63 0.915965594177219, bases.md:64 0.663700804613853
• bases.md:53-55 lattice sum 1.2851908043 + 1.51e-07 = 1.2851909555 against zeta(2) L = 1.2851909555, ten digits; the direct Dirichlet sum for bases.md:60 is 0.781302412896
• bases.md:67-68 Eisenstein sieve 311237/400000 = 0.77809 on 400000 pairs, against the predicted 0.778094
• bases.md:75-76 CF of L sqrt(3)/pi^2 is [0, 7, 3, 2, 2, 3, 2, 23, 2, 7, 1, 12, 1, 17, 1, 29, 1, 1, 2, 1, 6, 12], ordinary quotients, last convergent denominator 1805284405980
• bases.md:77 control zeta(2)/pi^2 has CF [0, 5, 1, 10^59], convergent 1/6
• bases.md:85 Cl2(pi/3) = 1.0149416064...
• bases.md:21 the figure files/figures/bases-fig.png; the run also reports 364/612 in-frame points visible = 0.5948

NOTE

• bases.md:76 states a bound, not a value: the printed last convergent denominator 1805284405980 is what clears 10^11.
• bases.md:173 claims byte-for-byte redrawing; the regenerated figure is pixel-identical to the recorded one but the PNG bytes differ.
• The recorded transcript labels the reflection residual ~1e-40 where the value is -1.1035169449471404E-42; bases.md:174 records that already, so the label is dropped here and the identity printed plainly.