Resonance-Weighted Metric Geometry in USP Field Theory: Effective Circumference Ratios, Not Variation of Mathematical Pi

This document develops a corrected and carefully bounded USP Field Theory interpretation of effective circumference ratios in resonance-weighted geometry. The central clarification is that the mathematical constant pi remains fixed. USP Field Theory does not claim that pi changes, and all USP Field Theory equations that use pi continue to treat it as the standard constant. Instead, this document defines operational quantities such as effective, coordinate-based, and proper metric circumference ratios that may differ from the Euclidean value when circumference and radius are measured through a non-Euclidean, frequency-dependent, or resonance-weighted metric. The paper introduces a resonance-weighted line element, distinguishes coordinate radius from proper metric radius, and separates coordinate circumference ratios from metric-consistent ratios. It also presents a weak-field USP metric ansatz using a normalized Delta-f mismatch proxy, explains why the leading scalar correction is quadratic under Delta-f sign symmetry, and adds a calibration pathway for the effective coefficient using analog optical systems such as gradient-index media or ring resonators. A connection to the weak-field isotropic form of the Schwarzschild metric is included as an alignment condition: USP must first reproduce the known conformal weak-field geometry before any residual interpretation can be considered. The document therefore functions as a mathematical and conceptual guardrail for future USP work involving resonance-weighted geometry, lensing diagnostics, ring-resonator analogs, and effective path-length measurements. This work is intended for understanding and formal clarification. It is not a claim that physical or mathematical pi varies. It provides a controlled language for cases where an effective measured circumference ratio becomes useful in mathematical modeling.