The Holographic-Refoundational Paradigm of Metrology: Integrating Classical and Quantum Inverse Methods

Inverse problems in physics and engineering are intrinsically ill-posed: classical solutions often exist only for idealized scenarios and become unstable under realistic conditions. For example, the electrostatic method of images – valid for a point charge above a grounded plane – fails for a charge near a dielectric sphere, requiring an infinite multipole expansion. Remarkably, in topological insulators (TIs) this classical failure is resolved: axion electrodynamics impose modified boundary conditions so that the unique solution is an image dyon (a combined electric q′ and magnetic p′ charge) instead of an infinite series. Similarly, analytical inversion formulas (e.g. for coil impedance in NDE) provide exact estimates of permittivity and conductivity, but are extremely sensitive to noise and model error. To overcome these fundamental instabilities, I propose Holographic–Homological Metrology, a unified paradigm. In this framework, a dense array of quantum sensors (e.g. NV-center magnetometers) creates a high-dimensional “hologram” of the sample’s boundary field. Persistent homology is then applied to extract Betti numbers – integer-valued topological invariants (connected components, loops, voids) – from this data manifold. These invariants are robust to noise and deformation and serve as stable signatures of the underlying state. Additionally, the classical analytical models are retained in a Differentiable Physics Oracle (DPO): physics laws (e.g. coil impedance relations) are encoded as soft constraints in machine-learning models. The result is a shift from fragile parameter estimation to reliable topological classification, with applications in non-destructive evaluation, biomedical diagnostics, and fundamental tests of topological matter.