Gauge Theoretic Signal Processing I: The Commutative Formalism for Single-Detector Adaptive Whitening

TL;DR

This paper introduces a geometric gauge-theoretic framework for adaptive whitening in gravitational-wave detectors, ensuring path-independent, stable, and causality-preserving noise filtering.

Contribution

It reformulates adaptive whitening as a geometric problem on a principal bundle, proving flatness and path-independence of the optimal filter update law.

Findings

Proves the curvature of the connection vanishes for scalar fields.

Establishes a holonomic, path-independent update law for filters.

Provides a rigorous foundation for real-time calibration stability.

Abstract

We present a geometric framework for adaptive whitening in gravitational-wave detectors, reformulating the problem from a sequence of spectral factorizations to parallel transport on a principal bundle. We identify the whitening filter as a section over the manifold of power spectra and derive the minimum-phase connection as the unique geometric structure that enforces signal causality while preserving signal-to-noise ratio. This construction yields a rigorous definition of geometric drift, a coordinate-independent scalar measuring the intrinsic instability of the detector noise floor. The central result is the flatness theorem, which proves that the curvature of the connection vanishes for scalar fields. This establishes a holonomic update law, guaranteeing that the optimal filter correction is path-independent and determined solely by the instantaneous noise state, free from geometric phase or hysteresis. This framework unifies the static theory of Wiener-Hopf factorization with the dynamic requirements of real-time control, providing a rigorous certification for the stability of zero-latency calibration routines and establishing a foundation for gauge-theoretic signal processing (GTSP) in next-generation detector networks.