= Solution
Combining <metric compatibility> with the <torsion-free> condition forces the <Koszul formula>:
$$
2g(\nabla_XY,Z)=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]).
$$
Nondegeneracy of $g$ proves uniqueness of the <Levi-Civita connection>. For existence, use the right side to define $2g(\nabla_XY,Z)$. Expanding brackets shows that this expression is $C^\infty$-linear in $Z$, so it defines a smooth one-form; the <musical isomorphism> gives the required <vector field>. The same expansion shows $\nabla_{aX}Y=a\nabla_XY$, additivity, and $\nabla_X(aY)=X(a)Y+a\nabla_XY$, establishing the connection rules. Subtracting the formulas with $X,Y$ exchanged gives $\nabla_XY-\nabla_YX=[X,Y]$. Adding the formulas pairing $\nabla_XY$ with $Z$ and $\nabla_XZ$ with $Y$ gives <metric compatibility>. Thus this construction has both required properties.
In coordinate <vector fields> the brackets vanish. Consequently the <Christoffel symbols> and coordinate <derivative> are
$$
\boxed{\Gamma^k_{ij}=\frac12g^{k\ell}(\partial_i g_{j\ell}+\partial_j g_{i\ell}-\partial_\ell g_{ij}),\qquad
\nabla_XY=\left(X^i\partial_iY^k+\Gamma^k_{ij}X^iY^j\right)\partial_k}.
$$
Repeated indices are summed. The displayed connection type in the source is best understood in its standard form on two <vector fields>; if the first input is an individual tangent vector at a point, the output lies in the tangent fiber at that point rather than in the space of global sections.
For the <parallel metrics with a common Levi-Civita connection>, let $\gamma$ be a piecewise smooth path from the point $x$ of equality to any $y$. <Connected> <smooth manifolds> admit such paths because coordinate balls are path <connected>. <Parallel transport> $P_\gamma$ for the common connection is invertible and preserves both metrics. Therefore
$$
\widetilde g_y(P_\gamma u,P_\gamma v)=\widetilde g_x(u,v)=g_x(u,v)=g_y(P_\gamma u,P_\gamma v).
$$
Every pair of tangent vectors at $y$ arises this way, so $\boxed{\widetilde g=g}$ everywhere.
Dropping the agreement at one point removes the conclusion. For any constant $c>0$, $\widetilde g=cg$ has the same <Christoffel symbols>. Nor must the two metrics be proportional: on $\mathbb R^n$, $n\geq2$, the constant metrics $\sum_i(dx^i)^2$ and $2(dx^1)^2+\sum_{i\geq2}(dx^i)^2$ both have zero connection coefficients. In general write $\widetilde g(u,v)=g(Au,v)$. Since both metrics are parallel, $0=(\nabla_X\widetilde g)(Y,Z)=g((\nabla_XA)Y,Z)$, so $\nabla A=0$. Conversely, a positive $g$-self-adjoint parallel $A$ makes the same <torsion-free> connection compatible with $\widetilde g$, proving equality of their <Levi-Civita connections>. Thus one-point agreement specifies $A=I$ and forces it everywhere; without it, nontrivial parallel choices can remain.
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