Zack: yeah, that's my mistake. I ran the code up till 997 and only built maximal progressions, so the 5-11-17 didn't get inserted until step 29, thanks to 5-11-17-23-29. I'll re-run once I fix up the code and edit the betti numbers tomorrow. Thanks!

The largest Eigenvalue is bounded by the Matrix norm: given your matrix $A$, the norm $\|A\|$ is defined as the supremum of $\frac{\|Av\|}{\|v\|}$ where $v$ ranges over unit vectors. You can see in the second section of [this paper][1] that any binary matrix $\|A\|$ is bounded above by $n\sqrt{n}$ where $n$ is the dimension of $A$. [1]: math.uconn.edu/~kconrad/blurbs/linmultialg/matrixnorm.pdf

Will: I see. Why do you think this would capture anything of interest in the present context? The second graph in the question that you linked to shows that the count of twin pseudoprimes doesn't even approximate the count of twin primes, and surely twin (blahs) is a degenerate simple case of (blahs) in arithmetic progression... Have I misunderstood?

Will: Sorry, but I don't know what Cramer Primes are, and googling has not yet helped. So the answer to your first question is certainly no. I'm not sure about connectedness (minus that pesky "2"), the only theorem I could think of using is Green Tao and that doesn't seem to help.

@David: you mean like this? arxiv.org/abs/1101.5704

Gerry: sadly, yes. I fear that my sense of humor is progressively deteriorating...

Dear Tim, I've made the edits. Thank you for your suggestion. Benjamin, I was aware of factor maps, but those are too specific since they require topological notions. In general, that framework wouldn't even accomodate shift maps on infinite strings in finitely many symbols since the space isn't compact, right?

Set $N = M$ and use $f = $ identity?

Professor Kalai: the "worst case" for that Voronoi idea is a simplex on n equidistant points, since all facets in sight have the same diameter as the original simplex. But in this case, we can just use the n vertices for generating the Voronoi diagram and no facet lies in the same Voronoi cell. Like I said before, it is a silly heuristic, but I can't seem to create a counterexample...

@Benjamin: sorry about that, I had maps between rings so "of course" they were ring homomorphism. Silly mistakes...