In 2017, Stefano Laporta published the complete four-loop contribution to the electron anomalous magnetic moment. The calculation — reduction of 891 Feynman diagrams to master integrals via integration-by-parts identities, followed by numerical evaluation using difference equations — produced the coefficient A₄ = −1. 91224576492645 to 1100 digits of precision. Most master integrals in the calculation were evaluated analytically: expressed as rational combinations of π, ζ (3), ζ (5), ln (2), and polylogarithms Liₙ (½). Six could not be. These six integrals, from topologies 81 and 83 in Laporta's classification, are known to 4925 digits but have no known closed form. The six integrals: C81a = +116. 6945857911866. . . C81b = −8. 7483203238146. . . C81c = −0. 2360852771203. . . C83a = +2. 7711919861455. . . C83b = −0. 8078473532638. . . C83c = −0. 4347026185438. . . For eight years, the multi-loop community — Broadhurst, Schnetz, Panzer, Brown, Adams, Bogner, Weinzierl, and others — has attempted to express these integrals in terms of known mathematical constants. Analytical methods (differential equations, sector decomposition, symbol methods, motivic approaches) have all failed for topologies 81 and 83 specifically. The integrals remain six numbers without names. This paper reports the results of a systematic numerical investigation: 24 PSLQ integer relation scans against the known transcendental basis, a complete cross-relation analysis proving mutual independence, and a sensitivity analysis showing that these constants contribute 43 times the measurement precision to the most precisely measured quantity in physics.
Geoffrey Howland (Wed,) studied this question.