= Solution
The <collocation order theorem> identifies the classical order with the order of its interpolatory quadrature. If $\sum_i b_i g(c_i)$ integrates every <polynomial> of degree at most $p-1$ exactly, but fails for some <polynomial> of degree $p$, then the collocation method has order $p$ for sufficiently smooth differential equations and the local implicit stage branch. With $s$ nodes, $s\le p\le2s$.
Equivalently, put $\pi_s(\tau)=\prod_i(\tau-c_i)$. If its first $k$ moments vanish,
$$
\int_0^1\tau^j\pi_s(\tau)d\tau=0\quad(0\le j<k),
$$
and the next does not, then $p=s+k$. <Polynomial> division explains the quadrature part: any <polynomial> through degree $s+k-1$ is a remainder of degree below $s$ plus $\pi_s$ times a <polynomial> of degree below $k$. The remainder is integrated exactly by interpolation, and the remaining integral vanishes. Conversely a nonzero next moment is an explicit failed quadrature test. The differential-equation order assertion is the collocation theorem being stated here. In particular <Gauss collocation methods> achieve $2s$, <Radau IIA methods> achieve $2s-1$, and endpoint-including <Lobatto IIIA methods> achieve $2s-2$ for $s\ge2$.
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