The Unifying Theory of Physics and Mathematics: Scale-Invariant Phase Rebound

Scale-Invariant Phase Rebound is a hypothesis about the books of a system at a bound. When a host theory drives recoverable content toward a threshold, that content is posted against the partner the host theory already possesses. The same turning is readable wherever a host theory will lend it names: in the Theory of Relativity as the Kruskal partner of a trapped congruence; on Riemann’s completed function ξ as the identification s ↦ 1 − s; in Ramanujan’s mock theta functions as Zwegers’s shadow; on a string worldsheet as T-duality and orientation; in supergravity as the fermionic partner of the graviton. This volume writes that hypothesis as a teaching of the Theory of Relativity and of the Hawking–Penrose incompleteness theorems, with the field equation, Bianchi identity, geodesic equation, Raychaudhuri equation, energy conditions, Bondi and Hawking masses, the Kruskal extension, and the official incompleteness hypotheses set out in lemmas. Recoverability of the exterior record is proved as Theorem GR-2 by exhausting the host partner list while the null energy condition remains in force. Ramanujan’s plane, the five degrees from Riemann to a worldsheet, an eleven-role counting of ten dimensions × two thermodynamic faces plus an observer, a working M-theory action, and supergravity as the void face of mass complete the costume. The work is offered to the physics and mathematics community as a finished teaching of the hypothesis.