An Information-Geometric Program for AM-GM Stability: Log-Sobolev Bounds and Variance Control

 This work proposes and develops an information-geometric program to establish a
 quantitative stability bound for the Arithmetic Mean– Geometric Mean (AM-GM)
 inequality De Ceuster [2025b]. We combine tools from information geometry, optimal
 transport, and functional inequalities (specifically the Log-Sobolev inequality, LSI).
 The core philosophy is to translate additive dispersion (variance σ2) into multiplicative
 deviation (log(A/G)) via probabilistic metrics. We formulate a sufficient-condition
 theorem: if an empirical distribution satisfies an LSI, an explicit exponential bound
 for the AM/GM ratio follows.