Maybe you prefer the statement "$Proj(R\otimes S)$ is the join of $Proj(R)$ and $Proj(S)$", where $R$ and $S$ are the monoid algebras, and "join" means the union of the lines through the two spaces?

I'd meant a cone on a square, in 3d. That cone has no edges of the special type I'm asking about. But if you want we can consider the 4d cone on a pyramid (with its base). That cone has 5 edges, of which only the line through the apex is special in the sense I'm asking about.

That's correct; I'm not asking about simplicial cones. Think instead of the cone on a square, i.e. a pyramid with no base.

+1 for the choice of book!

Off-topic, but the first spectral sequence I know in the literature is in A Christmas Carol.

Okay, sure. That's Schur-Weyl duality applied to #3. I don't know how to apply understanding of $S_n$ (or other finite group) representation theory to retrodict the L-R combinatorics, in anything analogous to the theory of crystals on the $GL(k)$ side (but would be very interested to hear).