What Is the Definition of Linear Predictive Coefficients When the Optimal Value Aren't Unique?
Constant zero sequence is the only finite sequence for which $a_i$ are not uniquely defined. 

Indeed, $a_i$ are a solution to the equation
$$
\begin{bmatrix}
R_0 &amp;&amp; R_1 &amp;&amp; \ldots &amp;&amp; R_{k-1} \\
R_1 &amp;&amp; R_0 &amp;&amp; \ldots &amp;&amp; R_{k-2} \\
\vdots &amp;&amp; \vdots &amp;&amp; \ddots &amp;&amp; \vdots \\
R_{k-1} &amp;&amp; R_{k-2} &amp;&amp; \ldots &amp;&amp; R_0
\end{bmatrix} 
\begin{bmatrix}
a_1 \\
a_2 \\
\vdots \\
a_k 
\end{bmatrix} = - 
\begin{bmatrix}
R_0 \\
R_1 \\
\vdots \\
R_{k-1} 
\end{bmatrix}
$$
To prove their uniqueness, it is sufficient to prove that the autocorrelation matrix R is nondegenerate.

If $y_0, y_1, \ldots, y_n$ is the initial sequence, $y_0 \ne 0$, and 
$$
Y = \begin{bmatrix}
y_0 &amp;&amp; \ldots &amp;&amp; y_n &amp;&amp; 0 &amp;&amp; \ldots &amp;&amp; 0 \\
0 &amp;&amp; y_0 &amp;&amp; \ldots &amp;&amp; y_n &amp;&amp; \ldots &amp;&amp; 0 \\
\vdots &amp;&amp; \ddots &amp;&amp; \ddots &amp;&amp; \vdots &amp;&amp; \ddots &amp;&amp; \vdots \\
0 &amp;&amp; \ldots &amp;&amp; 0 &amp;&amp; y_0 &amp;&amp; \ldots &amp;&amp; y_n
\end{bmatrix},
$$
then 
$$
R = YY^T,
$$
so R is the Gram matrix of the rows of Y. A Gram matrix is nondegenerate iff the vectors are linearly independent, which they obviously are.

To deal with the constant zeros problem, I should probably just trim silence at the beginning and end of the audio.