The Specular Mother Equation provides a unified informational framework in which quantum mechanics and general relativity emerge as complementary limits of a single dual-sheet dynamical structure. The theory is built on the extended manifold Mspec = M x H, combining a geometric sheet and a Hilbert sheet through the specular field Phi = (psi, g\ₘunu, I, theta). The equation couples geometric curvature, quantum evolution, informational flow, and phase dynamics through five core components: the specular curvature Kspec, the specular Hamiltonian Hspec, the extended derivative Vspec, the informational potential Vspec, and the informational boson Bspec (theta). Quantum mechanics arises in the non-collapsed regime where the informational boson is inactive and the dynamics reduce to Schrodinger-type evolution on the Hilbert sheet. General relativity emerges in the collapsed regime, triggered by phase activation near informational tilt-points, where bosonic amplification suppresses quantum contributions and induces effective geometric curvature. A stability theory based on the operator Lspec and its lowest eigenvalue Sspec characterizes transitions between regimes and explains observer-dependent geometric emergence. This work establishes the mathematical foundation of the Specular Framework and introduces the Specular Mother Equation as its central unifying structure, opening the way to multi-qubit extensions, cosmological applications, and the broader Specular Dynamics program.
Valentina Moroni (Fri,) studied this question.