= Solution
For a <smooth function> $f$ and the <Levi-Civita connection> $\nabla$, the <Riemannian Hessian> is the covariant two-tensor
$$
\operatorname{Hess}_g f(X,Y)=(\nabla_Xdf)(Y)=X(Yf)-df(\nabla_XY).
$$
The connection rules show that this is linear over <smooth functions> in both vector-field arguments, so it is a <tensor>. Torsion-freeness gives its symmetry:
$$
\operatorname{Hess}_g f(X,Y)-\operatorname{Hess}_g f(Y,X)
=[X,Y]f-df(\nabla_XY-\nabla_YX)=0.
$$
Its <metric trace> defines the paper's <Laplace-Beltrami operator>, $\Delta=\operatorname{div}_g\operatorname{grad}_g$; its <eigenvalues> on a <closed manifold> are nonpositive. We use $P=-\Delta$ when listing nonnegative <eigenvalues> or writing $e^{-tP}$.
At $p$, choose an <orthonormal basis> $e_1,\ldots,e_d$ of the <tangent space> and unit-speed <geodesics> $\gamma_i(t)=\exp_p(te_i)$. Differentiating twice along a curve gives
$$
\frac{d^2}{dt^2}f(\gamma_i(t))
=\operatorname{Hess}_g f(\dot\gamma_i,\dot\gamma_i)+df(\nabla_{\dot\gamma_i}\dot\gamma_i).
$$
The covariant acceleration vanishes on a <geodesic>. Taking the <trace> at $t=0$ proves the <geodesic trace formula for the Laplace-Beltrami operator>:
$$
\boxed{\Delta f(p)=\sum_{i=1}^d(f\circ\gamma_i)''(0)
=\lim_{h\to0}\frac{\sum_{i=1}^d[f(\exp_p(he_i))+f(\exp_p(-he_i))-2f(p)]}{h^2}.}
$$
The last equality is Taylor's formula along the <geodesics>, and expresses the <Laplacian> entirely through nearby function values.
For stabilization, let $X_1,X_2$ be the assumed isospectral nonisometric four-dimensional <tori>, and let $k=n-4>0$. Take the same small <flat torus> $F_\varepsilon=\mathbb R^k/\varepsilon\mathbb Z^k$ as an extra factor. On a <Riemannian product>,
$$
P_{X_i\times F_\varepsilon}=P_{X_i}\otimes I+I\otimes P_{F_\varepsilon}.
$$
If $\phi_j$ and $\psi_\ell$ are <eigenfunctions> with <eigenvalues> $\lambda_j$ and $\mu_\ell$, then $\phi_j\psi_\ell$ has <eigenvalue> $\lambda_j+\mu_\ell$. Products of <orthonormal eigenbases> are complete in the product $L^2$ space, so this lists all <eigenvalues> with <multiplicities>, including coincidences between different sums. Consequently the two products are isospectral.
One must also prove that multiplying by a common factor has not accidentally made the two <manifolds> <isometric>. Choose
$$
0<\varepsilon<\delta<\min\{\operatorname{inj}(X_1),\operatorname{inj}(X_2)\}.
$$
The injectivity radii are positive because the factors are <compact>. A <closed> product <geodesic> of length less than $\delta$ projects to a <closed geodesic> of length less than $\delta$ in $X_i$. Such a projected <geodesic> must be constant: otherwise its initial vector over one period would be a nonzero vector of norm below the <injectivity radius> mapping back to its starting point under the <exponential map>. Thus all sufficiently short <closed geodesics> lie in the $F_\varepsilon$ directions. Conversely the $k$ coordinate circles, of length $\varepsilon$, pass through every point and span those directions.
It follows that the <smooth distribution> tangent to the small flat-torus factor is characterized intrinsically by the span of tangent vectors to <closed geodesics> of length below $\delta$. A hypothetical product <isometry> would preserve this <smooth distribution> and its <orthogonal complement>. The latter has leaves $X_i\times\{q\}$; the <isometry> and its inverse take whole leaves to whole leaves, thereby giving an <isometry> $X_1\cong X_2$, a contradiction. This proves <isospectral stabilization by a small flat torus>:
$$
\boxed{X_1\times F_\varepsilon\text{ and }X_2\times F_\varepsilon\text{ are isospectral nonisometric }n\text{-tori for every }n>4.}
$$
This argument works even without assuming that the original four-dimensional <torus> metrics are flat.
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