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1576 bytes, as of the version from 2026-11-02 15:29 (current). Recorded here, not run.
"""Independent conifold sums from gamma-ratio asymptotics; heuristic, not certified.
c_n=P n^-e exp(sum L_k n^-k). The logarithmic coefficients are
Bernoulli-polynomial differences. Differentiating all upper and denominator
parameters together leaves e fixed and yields the coefficients of c_n h_n.
Each inverse-power tail is summed with Hurwitz zeta. Adapted from T679's
conifold period check, extended here to the first logarithmic solution.
"""
from mpmath import mp
def conifold_tail(alpha,beta=None,N=1400,K=130):
if beta is None:beta=[mp.mpf(1)]*len(alpha)
exponent=sum(beta)-sum(alpha)
L=[mp.mpf(0)];D=[mp.mpf(0)]
for k in range(1,K+1):
L.append((-1)**(k+1)*(sum(mp.bernpoly(k+1,a) for a in alpha)-sum(mp.bernpoly(k+1,b) for b in beta))/(mp.mpf(k)*(k+1)))
D.append((-1)**(k+1)*(sum(mp.bernpoly(k,a) for a in alpha)-sum(mp.bernpoly(k,b) for b in beta))/k)
E=[mp.mpf(1)]
for n in range(1,K+1):E.append(sum(k*L[k]*E[n-k] for k in range(1,n+1))/n)
H=sum(mp.digamma(b) for b in beta)-sum(mp.digamma(a) for a in alpha)
B=[H*E[n]+sum(D[k]*E[n-k] for k in range(1,n+1)) for n in range(K+1)]
c=mp.mpf(1);h=mp.mpf(0);F=mp.mpf(0);G=mp.mpf(0)
for n in range(N):
F+=c;G+=c*h
c*=mp.fprod(a+n for a in alpha)/mp.fprod(b+n for b in beta)
h+=sum(1/(a+n) for a in alpha)-sum(1/(b+n) for b in beta)
P=mp.fprod(mp.gamma(b) for b in beta)/mp.fprod(mp.gamma(a) for a in alpha)
Z=[mp.zeta(exponent+k,N) for k in range(K+1)]
return F+P*sum(e*z for e,z in zip(E,Z)),G+P*sum(b*z for b,z in zip(B,Z))