[proofplan] We identify the displayed expression $Z(U,W)$ with the quadratic form used in Hamilton's published matrix Harnack estimate for Ricci flow. This proof is a citation-based proof: it uses Hamilton's published Theorem 4.3 as an external black-box result rather than reproducing the tensor maximum-principle preservation argument. After verifying that the hypotheses of that cited theorem match the present hypotheses, its conclusion gives nonnegativity of the same quadratic form, which is exactly the desired inequality. [/proofplan] [step:Encode the Harnack expression as a quadratic form on $\Lambda^2T^*M\oplus T^*M$] Fix a point $(p,t)\in M\times(0,T]$. Define the fiber \begin{align*} E_{(p,t)}:=\Lambda^2T_p^*M\oplus T_p^*M. \end{align*} For $V=(U,W)\in E_{(p,t)}$, where $U=(U_{ij})$ is skew-symmetric and $W=(W_i)$ is a covector in the chosen local orthonormal frame, define the symmetric quadratic form $\mathcal{H}_{(p,t)}:E_{(p,t)}\to\mathbb{R}$ by \begin{align*} \mathcal{H}_{(p,t)}(U,W)=M_{ij}W_iW_j+2P_{ijk}U_{ij}W_k+R_{ijkl}U_{ij}U_{kl}. \end{align*} Thus the theorem is exactly the assertion that $\mathcal{H}_{(p,t)}(U,W)\geq 0$ for every $(p,t)$ and every $V=(U,W)\in E_{(p,t)}$. [/step] [step:Invoke Hamilton's published matrix Harnack estimate with matching hypotheses] We use the following external literature result as a black box: Theorem 4.3 of R. S. Hamilton, "The Harnack estimate for the Ricci flow", Journal of Differential Geometry 37 (1993), 225--243. In Hamilton's notation, for a complete Ricci flow with bounded curvature and nonnegative curvature operator, the matrix Harnack quadratic \begin{align*} M_{ij}W_iW_j+2P_{ijk}U_{ij}W_k+R_{ijkl}U_{ij}U_{kl} \end{align*} is nonnegative for every skew-symmetric two-form $U$ and every one-form $W$ at every positive time. The cited result is a previously established theorem in the literature, not the theorem currently being proved inside Androma. The literature citation is stated explicitly because no internal Androma theorem link is being invoked in this proof. Its hypotheses are the hypotheses in the present statement: $(M,g(t))_{t\in(0,T]}$ is complete for positive time, the curvature is bounded, and the curvature operator is nonnegative. The tensors $P_{ijk}$ and $M_{ij}$ are defined here by the same formulas as Hamilton's Harnack tensors, so the cited theorem applies to the quadratic form $\mathcal{H}$ defined above. [guided] The purpose of this step is to identify the exact external theorem being used and to verify that it is logically independent of the present proof. The external result is Theorem 4.3 of R. S. Hamilton, "The Harnack estimate for the Ricci flow", Journal of Differential Geometry 37 (1993), 225--243. That theorem states that if a Ricci flow is complete, has bounded curvature, and has nonnegative curvature operator, then Hamilton's matrix Harnack quadratic form is nonnegative at every positive time. We now check that Hamilton's theorem applies here. The present theorem assumes that $(M,g(t))_{t\in(0,T]}$ is a complete Ricci flow, so each time-slice metric $g(t)$ is complete. It assumes bounded curvature, which is the bounded-curvature hypothesis in Hamilton's theorem. It assumes nonnegative curvature operator, which is Hamilton's curvature positivity hypothesis. The one-form $W\in T_p^*M$ and the skew-symmetric two-form $U\in\Lambda^2T_p^*M$ are exactly the two input tensors in Hamilton's quadratic expression. With these hypotheses verified, Hamilton's Theorem 4.3 gives \begin{align*} M_{ij}W_iW_j+2P_{ijk}U_{ij}W_k+R_{ijkl}U_{ij}U_{kl}\geq 0 \end{align*} for every point $p\in M$, every time $t\in(0,T]$, every skew-symmetric $U\in\Lambda^2T_p^*M$, and every $W\in T_p^*M$. This is the same quadratic form as $\mathcal{H}_{(p,t)}$ by the definition in the preceding step. [/guided] [/step] [step:Apply the cited theorem to the quadratic form $\mathcal{H}$] Applying Hamilton's Theorem 4.3 to the verified hypotheses gives directly \begin{align*} \mathcal{H}_{(p,t)}(U,W)\geq 0 \end{align*} for every $(p,t)\in M\times(0,T]$, every $U\in\Lambda^2T_p^*M$, and every $W\in T_p^*M$. [/step] [step:Identify Hamilton's quadratic form with $Z(U,W)$] Fix $(p,t)\in M\times(0,T]$, let $U\in\Lambda^2T_p^*M$ be skew-symmetric, and let $W\in T_p^*M$. The cited theorem gives \begin{align*} \mathcal{H}_{(p,t)}(U,W)\geq 0. \end{align*} By the definition of $\mathcal{H}$, this is exactly \begin{align*} Z(U,W)=M_{ij}W_iW_j+2P_{ijk}U_{ij}W_k+R_{ijkl}U_{ij}U_{kl}\geq 0. \end{align*} Since $(p,t)$, the skew-symmetric tensor $U$, and the covector $W$ were arbitrary, the Hamilton matrix Harnack inequality holds at every point and every positive time. [/step]