Independently recomputed constants (mpmath dps=40)

All values below were recomputed tonight, not copied from secondary blogs.

Hexagonal packing (target)

• Center density δ_hex = 1/(2√3) = √3/6 = 0.2886751345948128822545743902509787278238
• Area density Δ_hex = π/(2√3) = 0.906899682117108915361219637392477477...
• Theorem 3.2 last-sign-change target: 2π r_*^2 = 4π/√3 = 7.255197456936871402376313030568622929136

Published Cohn–Elkies 2003 (Annals 157, 689–714)

• Table 3, n=2: best packing 0.28868, new upper bound 0.28868 (center density, five decimals).
• Table 4, n=2: 2π r^2 = 7.25520, forced double roots 21.77, 29.02, 50.79, 65.34, 90.19.
• Recomputed from the printed Table 4 R:
  • δ ≤ 7.25520/(8π) = 0.2886752357800797600176013700050665498581
  • ratio vs hex = 1.000000350516046419403644002640063972807

Levenshtein / CE Proposition 6.1 (exact Bessel)

• j_1 = first positive zero of J_1 = 3.831705970207512315614435886308160766565
• Δ ≤ j_1^2/16 = 0.9176231651327433285762361105393816868551
• δ ≤ j_1^2/(16π) = 0.2920885252530132815247208990820543499602
• ratio vs hex = 1.011824332092168126325936766864430803816

This certificate (certs/ce_d2_m5.json)

• R = 3627599/500000 = 7.255198
• δ ≤ R/(8π) = 0.2886751562026082140699334855631848636009
• ratio vs hex = 1.000000074851598708507619246976814816322
• Isolated last odd root of G: (1267758233/174737932, 878736008/121118135) ≈ (7.2551976464961250, 7.2551976464961259)
• Meets CE Table 3 (δ prints as 0.28868).
• Strictly below CE Table 4 printed R = 7.25520.
• Does not attain 4π/√3. Not a magic function.