Remark: Family 273. The entropy photon-number inequality

Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output s entropy photon number is at least the transmissivity-weighted average of the inputs. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.