Thermal balance and the amplifier entropy photon-number inequality

A variational proof candidate for the amplifier entropy photon-number inequality for two independent finite-energy multimode inputs, allowing correlations within either input group. The argument pairs signed logarithmic-mean interpolation with thermal relaxation, using stationary cancellation and a trace-norm Stein estimate to close the finite-energy argument. It also presents a Gaussian spectral refinement, a Gaussian equality characterization and a two-output correlation bound conditional on the general inequality. The Gaussian-input inequality itself is established prior work. The conjecture is due to De Palma, Mari and Giovannetti; the variational framework and four-weight interpolation are adapted from OpenAI's beam-splitter argument. The mathematical development and writing are AI-generated. Harjas Sandhu developed and coordinated the agentic system used for this investigation and is the repository record creator. Independent specialist review and priority assessment are pending. The manuscript page provides the full abstract, reader's explanation, provenance and citation information.