The Magnet Is the Witness of the Pair Correlation: Time Reversal Broken by a Magnetic Term Turns Orthogonal Level Statistics into the Unitary Class of the Riemann Zeros

Montgomery found that the Riemann zeros repel one another with the pair correlation of the Gaussian unitary ensemble, and Odlyzko confirmed it to high precision far up the line. The unitary class is the class of quantum systems whose time reversal is broken; systems that keep time reversal fall in the orthogonal class. Time reversal acts on a two-level Hermitian block as complex conjugation, its fixed set is the real blocks, and a magnetic, time-odd term is exactly what leaves that set; degeneracy then costs three conditions instead of two, and levels repel as the square of their spacing instead of linearly. The two classes are constructed here as tridiagonal beta-ensembles and measured: spacing variance 0.287 for the orthogonal class and 0.177 for the unitary class, against exact values 0.286 and 0.180, with an uncorrelated control at 0.983. The witness of the zeros' class is therefore the magnet, the time-odd, priced branch of the arrow, and not the nucleus. This corrects the program's earlier census, which named nuclear spacings as the carrier: nuclei keep time reversal and follow the orthogonal class. The zeros' statistics carry the arrow in them.

Version 1.4.0, single-column Math Journal edition. The Lean engines and Fortran twins are printed in the appendices with the compiler's transcript; the Markdown source is in git, 1000sapients/Trisduction, protocols/RH One Bit/Nothing Escapes Twenty Three Rows/Witnesses/.