= Solution
Let $A$ be the interior-grid <matrix> representing $\Gamma_9$, with zero boundary error. It is real symmetric and has the stated <sine> eigenbasis. Put $s=\sin^2(k\pi h/2)$ and $t=\sin^2(\ell\pi h/2)$; the negative of its <eigenvalue> is
$$
-\lambda_{k\ell}=4s+4t-\frac83st.
$$
This increases in each variable on $[0,1]^2$. With $s_1=\sin^2(\pi h/2)\geq h^2$, the smallest absolute <eigenvalue> is at least $8s_1-(8/3)s_1^2\geq(16/3)h^2$. Hence its <Euclidean norm> <operator norm> satisfies
$$
\|A^{-1}\|_2\leq\frac3{16h^2}.
$$
The corrected scheme has an unscaled residual $r$ whose individual entries are uniformly $O(h^6)$. There are $m^2$ entries and $m<h^{-1}$, so $\|r\|_2=O(h^5)$. Subtracting the exact nodal equations from the numerical equations gives $Ae=-r$ (with sign depending on the residual convention), and therefore
$$
\boxed{\|e\|_2\leq\|A^{-1}\|_2\|r\|_2=O(h^3).}
$$
Choose a constant slightly larger than the resulting bound to obtain the requested strict $\|e\|<ch^3$ for all sufficiently small $h$. Uniform smoothness is needed for an $h$-independent constant; unspecified smoothness only in the interior would not by itself guarantee it.
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