A dimensionless Information–Energy–Entropy efficiency index (V‑index): definition, invariants, and Landauer‑linked validation

This paper introduces the V-index, a novel dimensionless metric designed to quantify efficiency at the intersection of information theory and thermodynamics. The index integrates three fundamental pillars: Shannon self-information (surprisal), the energetic cost of physical operations (mapped via the Landauer principle), and the resulting entropy change. The V-index provides a unit-consistent ratio that allows for the direct comparison of information processing efficiency across diverse physical platforms, from classical to mesoscopic systems. I provide a formal proof showing that under Landauer-optimal, quasi-static erasure conditions, the index converges to a universal invariant value of exactly two. The proposed scaling laws are tested against high-precision experimental data, including classical colloidal one-bit memories (Nature, 2012) and nanomagnetic memory bits (Science Advances, 2016). The work defines specific scaling behaviors, such as the linear energy multiplier law for supra-minimal dissipation and the hyperbolic law for fractional information erasure. By unifying energy, information, and entropy into a single dimensionless number, the V-index serves as both a practical benchmarking tool for future computing technologies and a potential candidate for a general physical law governing information-thermodynamic efficiency.