The Edge Run
Before it is terrain and before it is a river, it is this: the left edge of rule 30, in the frame that travels with it. The lit band along the top is the run of ones, and its depth at each column is the whole of what Ruler Range and Ruler Rivers are built from.
The edge
One cell wide at the start, evolved by
E ← E ⊕ ((E<<1) | (E<<2)). Depth runs downward,
generation runs to the right. Drag it.
The law, and why it is not a coincidence
Bit j of the edge is exactly periodic, with period 2p(j). Every bit is a clock. The run of ones at column g is simply how many of those clocks strike midnight together:
run(g) = #{ b : 2p(b) divides g+1 }
So it depends only on v₂(g+1) — how divisible the column
index is by two. Nothing about the run needs the automaton once you know that, which
is why the other two pages never evolve a single generation. The counter above checks
the shortcut against the real thing, live, on whatever window you are looking at.
The ladder
Group the columns by v₂(g+1) and the run takes one value per
group. That list is the whole generator.
Read the gaps. A plain ruler sequence climbs by one every step. This one jumps +6, +8, +5 — three whole scale classes out of nowhere, from the spectrum's own doubling plateaus. In Range those are the abrupt mountain classes; in Rivers they are confluences where the river suddenly gets much bigger. Noise has no reason to do either.
Where it goes: Ruler Range → · Ruler Rivers → · Ruler Flow →
p(j) is the 43-term verified spectrum from prefixlaw; past
v₂(g+1) = 17 the run leaves the verified range and those columns are
marked rather than guessed. The picture here is evolved honestly, one generation at a
time, purely so the shortcut has something to be checked against — it is the only
page in this set that does any work per column.