The Schrödinger Equation as a Resolution Geometry Theorem: Deriving Quantum Mechanics as Phase Tension Preservation on a Complex Scaffold

This companion paper demonstrates that the Schrödinger equation is a geometric inevitability: it is the unique dynamics permitted for a complex-valued ledger that must preserve probability amplitude while minimizing tension.

While the heat equation describes the relaxation of a scalar magnitude (dissipation), the Schrödinger equation describes the evolution of a magnitude-plus-phase ledger (rotation). In Resolution Geometry terms, the imaginary unit i acts as the Layer 2 rotation operator, converting spatial gradient tension not into decay, but into temporal frequency.

We derive the Schrödinger equation as the Euler-Lagrange condition for maintaining a smooth phase field on a 2D scaffold under the constraint of norm conservation. The resulting dynamics preserve the total 'phase tension' (energy) by converting it into unitary propagation. This completes the 'Resolution Geometry Trilogy,' demonstrating that Black-Scholes (finance), the Heat Equation (thermodynamics), and Schrödinger (quantum mechanics) are three expressions of the same constraint geometry—distinguished only by whether their ledger is scalar or complex, and whether their constraint is dissipative or unitary.