scalar_certificate.json

{
  "source": {
    "author": "Stefan Steinerberger",
    "title": "On a problem involving unit fractions",
    "version": "arXiv:2403.17041v5",
    "version_date": "2024-04-28",
    "page": 3
  },
  "parameter": {
    "c_rational": "76847/2000000",
    "c_source_decimal": "0.0384235",
    "logarithm_terms": 40,
    "exponential_terms": 65,
    "outward_mesh_denominator": 1000000000000000
  },
  "formulas": {
    "z": "(y-1)/(y+1), y>=1",
    "L": "2*sum(j=0..39,z^(2j+1)/(2j+1))",
    "U": "L+2*z^81/(81*(1-z^2))",
    "a_bounds": "4*[L(2),U(2)]+[L(1/(16c)),U(1/(16c))]-2",
    "T": "sum(j=0..64,4^j/j!)",
    "W": "T+(4^65/65!)/(1-4/66)",
    "Q_bounds": "[(1+c^2*T)/2,(1+c^2*W)/2]",
    "F_lower": "-c*a_upper^2/2-c*U(1/Q_lower)",
    "F_upper": "-c*a_lower^2/2-c*L(1/Q_upper)"
  },
  "enclosures": {
    "log2": {
      "lower_numerator": 693147180559945,
      "upper_numerator": 693147180559946,
      "common_denominator": 1000000000000000
    },
    "a": {
      "lower_numerator": 1259086027404675,
      "upper_numerator": 1259086027404676,
      "common_denominator": 1000000000000000
    },
    "exp4": {
      "lower_numerator": 54598150033144239,
      "upper_numerator": 54598150033144240,
      "common_denominator": 1000000000000000
    },
    "f": {
      "lower_numerator": -54110786906961,
      "upper_numerator": -54110786906960,
      "common_denominator": 1000000000000000
    }
  },
  "strict_checks": {
    "a_positive": true,
    "Q_inside_open_unit_interval": true,
    "f_less_than_minus_0_0541": true,
    "0_054_greater_than_0_07_log2": true
  },
  "explanation": "theorem.md, finite enclosure argument (1)-(6)",
  "scope": "The fixed scalar inequalities only. The integer cutoff and eventual limit are proved in theorem.md. No global optimization, explicit n0, formal verification, or independent review is asserted by this data."
}