Ignoring certificates preserves polynomial time

For every $\chi:B^*\to B$ and every hypothesis $D(\chi)$, the pair function $R(w,y)=\chi(w)$ satisfies $C(R)$. Thus the asserted machine must compute the first-word Boolean value on every tagged pair $(w,y)$, with a polynomial bound in $|w|+|y|$, including arbitrary or empty second words; the second word need not satisfy a certificate-length constraint. Here $B=\{\mathrm{false},\mathrm{true}\}$, $B^*$ is the set of all finite Boolean lists, including the empty list, and $|w|$ is list length. Write $D(\chi)$ for existence of such a machine and a polynomial $p\in\mathbb N[X]$ that, for every $w\in B^*$, compute the singleton output $[\chi(w)]$ from input $w$ in at most $p(|w|)$ steps. Write $C(R)$ for existence of such a machine and a polynomial $p\in\mathbb N[X]$ that, for all $w,y\in B^*$, compute $[R(w,y)]$ in at most $p(|w|+|y|)$ steps from the list obtained by tagging every bit of $w$ with the left injection into $B\sqcup B$, tagging every bit of $y$ with the right injection, and concatenating those two lists. A machine in these assertions is a Mathlib TM2 stack machine with finitely many stack indices, instruction labels, and control states, a finite input-stack alphabet, designated input and output stacks, a program, and initial label and control state; its other stack alphabets need not be finite. Input and output alphabet bijections transport the specified encoded lists to the corresponding stack alphabets. Computation starts with only the input stack populated, and reaches a halted configuration with the specified output on the output stack, all other stacks empty, and the control state reset to its initial value. Time counts executions of whole TM2 statements, each of which may contain several stack operations. The supplied body is admitted with sorry; no proof of this assertion is supplied there.