Remark: Family 194. Lech's multiplicity conjecture

Proves $e(R)\le e(S)$ for every flat local homomorphism of nonzero Noetherian local rings, where $e$ is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.

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.