Black-Scholes as a Resolution Geometry Theorem: Deriving Option Pricing from Membrane Tension Minimization

This companion paper demonstrates that the Black-Scholes partial differential equation emerges naturally from Resolution Geometry's framework of membrane tension minimization. The (S, t) price-time plane functions as a 2D scaffold; the option price surface V(S,t) is a fold over this scaffold; and the Black-Scholes PDE is the constrained tension-minimizing evolution (a gradient flow) for this surface under no-arbitrage conditions. The key insight is that the option price surface behaves like a soap film (minimal surface), not a stiff plate. The system minimizes Delta-squared (gradient/tension), and Gamma (curvature) emerges as the reaction force. By transforming to logarithmic coordinates—the 'fundamental scaffold' where the geometry is flat—the Black-Scholes equation reveals itself as pure diffusion with drift, with all metric corrections vanishing. This mapping suggests that financial derivatives pricing and gravitational physics share a common mathematical substrate: both are optimization problems on 2D manifolds with finite distinguishability capacity. Resolution Geometry is therefore not merely a framework for physics, but for any system characterized by finite capacity, conservation requirements, and cost minimization dynamics.