Resonant Geometry: π as a Field-Dependent Constant

This work introduces a resonant geometric framework where π emerges as a dynamic ratio determined by the local resonance field n(x,y). Using Riemannian notation, the metric is extended to ds²_φ = n²(x,y)(dx² + dy²), linking curvature directly to Δf² field tension.

The study defines π_res = π·n(r)/(1/r ∫₀ʳ n(ρ)dρ) and demonstrates how field inhomogeneities or Δf gradients modify π slightly, producing measurable deviations near strong-resonance zones.

Observational predictions suggest minor π_res shifts inside protons (Δf ≈ 10²³ Hz), near galactic centers, and in early-universe relic curvature.

This document (msf:45708) forms part of the USP Field Theory series and unifies resonance-based curvature with Riemannian geometry.