# Author: David Niedbala Giraudin (ORCID 0009-0009-1526-1178), independent researcher, France.
# Ancillary file of the note 'The Schur-Siegel-Smyth trace problem: the lower bound 1.8022246'.
# Written with the assistance of an AI system (Claude, Anthropic).
# Independent non-rigorous cross-check: F at chosen points by direct numerical integration (mpmath),
# with g read from the certificate as a piecewise-linear density (no closed-form potential formulas).
import json, sys, mpmath as mp
from fractions import Fraction as Fr
mp.mp.dps = 40
c = json.load(open(sys.argv[1]))
e = [mp.mpf(Fr(x).numerator)/Fr(x).denominator for x in c['nodes']]
rho = [mp.mpf(Fr(x).numerator)/Fr(x).denominator for x in c['rho']]
lamQ = [mp.mpf(Fr(x).numerator)/Fr(x).denominator for x in c['lamQ']]
lam = mp.mpf(Fr(c['lambda']).numerator)/Fr(c['lambda']).denominator
IS = mp.mpf(sys.argv[2])     # I(g)/s from the rigorous checker
N = len(e) - 1
gv = [mp.mpf(0)]*(N+1)
for k in range(1, N): gv[k] = rho[k-1]*2/(e[k+1]-e[k-1])
s = sum(rho)
def Ug(x):
    tot = mp.mpf(0)
    for k in range(N):
        a, b = e[k], e[k+1]
        if gv[k] == 0 and gv[k+1] == 0: continue
        # integrate in t = y - x so that the singularity sits at t = 0 exactly
        f = lambda t: mp.log(abs(t))*(gv[k]+(gv[k+1]-gv[k])*(x+t-a)/(b-a))
        if a < x < b: tot += mp.quad(f, [a-x, 0, b-x])
        else: tot += mp.quad(f, [a-x, b-x])
    return tot
for xs in sys.argv[3:]:
    x = mp.mpf(xs)
    val = x - lam + IS - 2*Ug(x)
    for Q, l in zip(c['A'], lamQ):
        if l: val -= l*mp.log(abs(mp.polyval([mp.mpf(q) for q in Q], x)))
    print(f"F({xs}) = {mp.nstr(val, 12)}", flush=True)
print("mass s =", mp.nstr(s, 15))