Quantization of Physical Interaction Strengths via Singular Moduli

Abstract

Since the 2019 redefinition of the SI units, precision metrology has sought to anchor all physical quantities to fundamental constants and integer invariants. While the optical frequency comb revolutionized timekeeping by discretizing the continuum of light into countable teeth, and the Quantum Hall Effect standardized resistance via topological invariants, a comparable standard for interaction strength remains elusive. Currently, measuring the coupling constant ($g$) between quantum systems is an estimation problem, inherently subject to drift, noise, and fabrication variance. Here, we introduce Interaction Metrology, a protocol that transforms the measurement of coupling strengths from an analog estimation into a topological counting problem. By engineering a specific class of algebraic catastrophe -- the Unimodal $X_9$ singularity -- in a driven-dissipative lattice, we prove that the system's interaction moduli are topologically forced to take discrete, quantized values, forming a "Geometric $k$-Comb." We derive the universality class of this quantization, showing that it arises from the discrepancy between the Milnor ($μ$) and Tjurina ($τ$) numbers of the effective potential, a strictly non-Hermitian effect forbidden in standard quantum mechanics. Finally, we provide an ab-initio blueprint for a silicon nitride implementation, demonstrating that this quantization is robust against disorder levels exceeding current foundry tolerances. This discovery establishes a universal standard for force sensing and quantum logic gates, enabling the calibration of interaction strengths with topological certainty.