= Solution
Let $M$ be a <closed manifold> with a <Riemannian metric> and use $\Delta=-\operatorname{div}\operatorname{grad}$, the <positive Laplace-Beltrami operator>. This is the setting in which the final discrete <eigenfunction expansion> applies without additional boundary or noncompact spectral hypotheses. A <Riemannian heat kernel> is a smooth $H(t,x,y)$ for $t>0$ such that
$$
(\partial_t+\Delta_x)H=0,\qquad
u(t,x)=\int_M H(t,x,y)f(y)\,dV_y\longrightarrow f(x)\quad(t\downarrow0)
$$
for every smooth initial function $f$. Here $\nu$ solves the <heat equation>. The limit is the statement that the initial kernel is the <Dirac delta distribution> on the diagonal. In the closed setting this uniquely determines the <heat kernel>; it is symmetric, preserves constants, is nonnegative, and satisfies the <semigroup property>.
A <heat parametrix> $P(t,x,y)$ is an approximate version with the same delta initial limit and with error
$$
R=(\partial_t+\Delta_x)P
$$
regular enough at $t=0$ to correct by a convergent integral series. One may construct it by the local Gaussian ansatz
$$
P(t,x,y)=\chi(x,y)(4\pi t)^{-m/2}e^{-d(x,y)^2/(4t)}\sum_{j=0}^{q}t^j u_j(x,y),
$$
where $\chi=1$ near the diagonal and is supported in a <convex normal neighborhood>, and the smooth coefficients solve the usual radial transport equations. The leading coefficient accounts for the <Riemannian volume form>. Taking $q$ sufficiently large makes the residual extend continuously, with any prescribed finite number of derivatives, to $t=0$. Alternatively a smooth asymptotic summation of all transport coefficients gives a residual vanishing to every order. These local construction, extension and differentiability facts are used here as subsidiary results, as permitted.
For the correction proof take a <heat parametrix> whose residual is smooth up to $t=0$ on a short interval $[0,T]$. We also use its uniform integral bound $\sup_{0<t\le T,x}\int_M|P(t,x,y)|\,dV_y<\infty$, its approximate-identity limit, and the following standard differentiability property: convolution with a smooth time-dependent kernel gives a smooth kernel for $t>0$, and differentiating it yields the identity below. These properties follow from the Gaussian estimates and the delta initial limit; no conclusion about the final exact <heat kernel> is assumed.
Define the <Volterra convolution of kernels> by
$$
(A*B)(t,x,y)=\int_0^t\int_M A(t-s,x,z)B(s,z,y)\,dV_z\,ds.
$$
It is associative wherever these integrals converge. The delta term at the upper endpoint gives
$$
(\partial_t+\Delta_x)(P*B)=B+R*B.
$$
Seek $H=P+P*Q$. Then its error vanishes exactly when $Q+R+R*Q=0$. The solution is the <Volterra parametrix correction>
$$
Q=\sum_{k=1}^{\infty}(-1)^kR^{*k},\qquad H=P+P*Q.
$$
The signs matter: the first correction is $-P*R$.
To prove convergence, let $|R|\le C$ and $V=\operatorname{vol}(M)$. The time variables in $R^{*k}$ range over a simplex of volume $t^{k-1}/(k-1)!$, so
$$
|R^{*k}(t,x,y)|\le C^k V^{k-1}\frac{t^{k-1}}{(k-1)!}.
$$
The series for $Q$ therefore converges uniformly on $[0,T]\times M\times M$. Derivatives obey analogous bounds, with polynomial factors in $k$, by the quoted residual extension and differentiation properties. Thus the series can be convolved and differentiated as above. Associativity and absolute convergence give $Q+R+R*Q=0$, proving $(\partial_t+\Delta_x)H=0$. The correction $P*Q$ has integral norm $O(t)$ by the integral bound on $P$ and boundedness of $Q$. It has zero initial limit, so $H$ has precisely the required delta initial data.
For uniqueness, a smooth solution $w$ with zero initial data satisfies the <heat equation energy identity>
$$
\frac{d}{dt}\|w(t)\|_{L^2}^2=-2\langle w,\Delta w\rangle=-2\|dw\|_{L^2}^2\le0.
$$
Its initial norm is zero, hence $w=0$. Applying uniqueness to consecutive evolutions proves the <semigroup property>. Choose $s_0<T$ and, for arbitrary $t>0$, compose enough kernels $H(t/r)$ that $t/r<s_0$. The resulting kernel is independent of the subdivision by uniqueness, is smooth for positive time by the quoted differentiability property, and extends the construction to every $t>0$. This proves \b[a parametrix determines the global heat kernel] in the closed setting. The <heat equation maximum principle> gives nonnegativity and applying uniqueness to the constant initial function gives conservation of total mass. Self-adjointness of the <positive Laplace-Beltrami operator> $\Delta$ gives symmetry.
Finally let $\{\phi_j\}$ be a complete complex <orthonormal eigenbasis> of $\Delta$, with <eigenvalues> $\lambda_j\ge0$, repeated by multiplicity. As subsidiary analytic facts we use the compact elliptic <compact elliptic spectral theorem>, <elliptic regularity> bounds making each fixed derivative of $\phi_j$ grow at most polynomially in $1+\lambda_j$, and a polynomial eigenvalue-counting bound. The <spectral expansion of the Riemannian heat kernel> converges because exponential decay makes the following series converge with every derivative when $t\ge\varepsilon>0$:
$$
\boxed{H(t,x,y)=\sum_{j=0}^{\infty}e^{-t\lambda_j}\phi_j(x)\overline{\phi_j(y)}.}
$$
Indeed, evolving initial data $\phi_j$ gives $e^{-t\lambda_j}\phi_j$ by direct substitution into the <heat equation> and uniqueness. For general $f\in L^2(M)$, completeness and the <heat equation energy identity> give
$$
\nu(t,x)=\sum_j e^{-t\lambda_j}\langle f,\phi_j\rangle\phi_j(x).
$$
The smoothly convergent kernel series represents this same operator, and equality for all smooth $f$ identifies it pointwise with the constructed <heat kernel>. The complex conjugate is required by the <inner product>; for a real <eigenbasis> it may be omitted. A noncompact <Riemannian manifold> may instead require a spectral integral, and the unqualified discrete formula should not be asserted there.
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