"The Schrödinger Equation from Gravitational Null-Zone Geometry

This is the third paper in the Eigenfield series. Building directly on the original framework (Zenodo 18646417, v11) and its companion note on the derivation of ℏ, we show that the time-symmetric nature of gravity at null zones (saddle points where net gravitational force vanishes) naturally forces the Schrödinger equation to appear — without any external quantum postulate.

The central insight is gentle and geometric: for the retarded + advanced balance condition \(\vec{g}_{\rm ret} + \vec{g}_{\rm adv} = \vec{0}\) to hold at every instant, the residual tidal field can only vibrate in closed orbits on the eigenvector sphere \(S^2\). Open or fractional orbits eventually break the saddle condition. Closed orbits require the action to come in exact integer multiples of a fundamental unit — the same unit we previously identified as Planck’s reduced constant \(\hbar = c^5 / (16\pi G \delta\lambda_{\rm min})\).

From this single requirement we derive the full harmonic-oscillator Schrödinger equation for a test particle near a null zone. Every term (potential, frequency \(\omega = \sqrt{\delta\lambda_{\rm min}}\), and \(\hbar\)) arises directly from the tidal residual and time-symmetric propagation at \(c\).

Written at a level accessible to advanced undergraduates while remaining mathematically precise, the paper uses only the eigenvalue algebra, interference diagnostics, and torque/precession machinery already present in the earlier work. No new postulates are introduced — quantisation emerges as a boundary condition imposed by the geometry itself.

This provides a concrete, pedagogical bridge from classical multi-body gravitational geometry to the form of quantum mechanics.