The Laporta Constants

The electron anomalous magnetic moment aₑ has been measured to 13 significant digits. The measurement — one electron trapped in a Penning trap at Harvard, its spin precession compared to its orbital frequency — produces: aₑ (exp) = 0. 00115965218059 (13) This number is predicted by quantum electrodynamics as a power series in the fine structure constant: aₑ = A₁ (α/π) + A₂ (α/π) ² + A₃ (α/π) ³ + A₄ (α/π) ⁴ + A₅ (α/π) ⁵ +. . . The coefficients A₁ through A₅ are computed from Feynman diagrams. A₁ = 1/2 (Schwinger, 1948, one diagram). A₂ = −0. 3285. . . (Petermann, Sommerfield, 1957, seven diagrams). A₃ involves 72 diagrams. A₄ involves 891 diagrams. A₅ involves 12, 672 diagrams. A₁ through A₃ are known analytically — every term expressed in closed form using rational numbers, π, ζ (3), ζ (5), ln (2), and polylogarithms. A₅ is known only numerically to moderate precision. A₄ sits between. In 2017, Stefano Laporta published the complete four-loop calculation. He evaluated all 891 diagrams numerically to extraordinary precision — 4800+ digits. The calculation reduced the diagrams to master integrals using integration-by-parts identities. Most master integrals were evaluated analytically. Six could not be. Those six master integrals — labeled C81a, C81b, C81c, C83a, C83b, C83c from their topology numbers in Laporta's classification — are known to 4925 digits but have no known closed form. For eight years, the multi-loop community has attempted to express them in terms of known mathematical constants. No one has succeeded. This paper asks: are these six numbers expressible in terms of known constants, or are they genuinely new?