Theorem: Exact finite certificate for the constant 1/5

The scalar constraints

imply $\eta \ge 1/5$.

The numerical proof requires only the rational identity

on $0\le e\le 1/10$, together with a split at $e=1/10$.

The constant $1/5$ is optimal for this scalar relaxation:

satisfies all four scalar constraints. This does not assert attainability by a physical state or protocol.

Proof. For $e\ge 1/10$, use $\eta \ge 2e$. Otherwise the two factors on the right of (eq:rational-certificate) are nonnegative. Thus $f\le 3/10+1/10=2/5$, and $\eta \ge 1-2f\ge 1/5$. The displayed equality tuple proves the relaxation statement.