  We construct $n$-node graphs on which any $O(n)$-size spanner has additive
error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound
of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes
the first two steps of a particular three-step research program, introduced in
prior work and overviewed here, aimed at producing tight bounds for the problem
by aligning aspects of the upper and lower bound constructions. More
specifically, we develop techniques that enable the use of inner graphs in the
lower bound framework whose technical properties are provably tight with the
corresponding assumptions made in the upper bounds. As an additional
application of our techniques, we improve the corresponding lower bound for
$O(n)$-size additive emulators to $+\Omega(n^{1/14})$.
