"""Certify Gram arguments for numberdb.org/T586.

Root routine reused from the live T583 generator, fetched 2026-10-28.
Analytic log-gamma fixes the phase continuously. Newton only proposes a
candidate; strict endpoint signs certify its enclosure. See certification.md.
"""
import os
import sys
import re
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField

WORKING_GUARD = 160
ROOT_GUARD_DIGITS = 10


def phase(t):
    """Continuous Riemann-Siegel phase, using analytic log-gamma."""
    R = t.parent()
    C = ComplexBallField(R.precision())
    return C(QQ(1)/4, t/2).log_gamma().imag() - t*R.pi().log()/2


def phase_derivative(t):
    R = t.parent()
    C = ComplexBallField(R.precision())
    return (C(QQ(1)/4, t/2).psi().real() - R.pi().log())/2


def gram_point(n, digits=100):
    if not isinstance(n, int) or n < -1:
        raise ValueError('The phase index must be an integer at least -1')
    R = RealBallField(numberdb.bits(digits, losing=WORKING_GUARD))
    assert phase_derivative(R(7)) > 0
    assert phase(R(7)) < -R.pi()
    target = n*R.pi()
    t = R(10 + 6*(n+1))
    # No claim that a Newton candidate is an enclosure of the root.
    for _ in range(20):
        t_next = R((t - (phase(t)-target)/phase_derivative(t)).mid())
        if t_next == t:
            break
        t = t_next
    center = t.mid().exact_rational()
    radius = QQ(1)/10**(digits+ROOT_GUARD_DIGITS)
    left, right = center-radius, center+radius
    assert left > 7
    low_value = phase(R(left)) - target
    high_value = phase(R(right)) - target
    assert low_value.is_finite() and high_value.is_finite()
    assert low_value < 0 < high_value, ('failed endpoint certificate', n)
    ball = R(left).union(R(right))
    assert ball.is_finite() and phase_derivative(ball) > 0
    assert ball.accuracy() >= numberdb.bits(digits)
    return ball