Size Ramsey number of star forests: prove $\hat{R}(F_1,F_2)=\sum_k\max\{n_i+m_j-1\}$ (Erdős #561)

The (asymmetric) size Ramsey number $\hat{R}(F_1,F_2)$ is the least number of edges $m$ such that there exists a graph $H$ with $m$ edges for which every red/blue colouring of the edges of $H$ contains a red copy of $F_1$ or a blue copy of $F_2$. Let $F_1=\bigcup_{i\leq s}K_{1,n_i}$ and $F_2=\bigcup_{j\leq t}K_{1,m_j}$ be star forests (disjoint unions of stars), with $n_1\geq\cdots\geq n_s\geq 1$ and $m_1\geq\cdots\geq m_t\geq 1$. Prove that $$\hat{R}(F_1,F_2)=\sum_{2\leq k\leq s+t} l_k,\qquad l_k=\max\{\,n_i+m_j-1 : i+j=k\,\}.$$

Acceptance. FULLY RESOLVES: a complete proof of the identity $\hat{R}(F_1,F_2)=\sum_{2\leq k\leq s+t} l_k$ for all star forests $F_1,F_2$ — machine-checkable (Lean/Coq) preferred, else a full written proof establishing both the upper bound (an explicit host graph with $\sum_k l_k$ edges that forces a monochromatic copy) and the matching lower bound. ADVANCES: prove the identity for a new family of parameter patterns that strictly generalizes the Győri–Schelp condition or the Davoodi–Javadi–Kamranian–Raeisi cases (state precisely which patterns and why they are new), with proof; OR verify the identity computationally for all star forests with $s+t\leq N$ and all part sizes $\leq M$, via explicit host construction plus an exhaustive-colouring certificate, reporting the attained $(N,M)$. Deliver the proof file, the new-case proof, or the verification code plus certified ranges.