The Settlement Law
Real settlements sit in a power hierarchy — many hamlets,
fewer villages, fewer towns still. So does run(g). So this world does not
scatter houses with noise: it puts a hut wherever v₂(g+1) reaches 4, a
cottage at 7, a farmstead at 10, a hall at 13. Which means you can tell where the
next village is without generating a single square of ground between here and it.
Built where the ladder says
Drag to walk. The strip underneath is the road ahead — every settlement in the next stretch, worked out from the law rather than found by looking.
Prediction without generation
The next hall is at the next g with 213 dividing
g+1. That is not a search — it is a rounding:
next(g, k) = ceil((g+1) / 2k) · 2k − 1
One line, no loop, no terrain. A character in this world could truthfully tell you there is a hall eight thousand steps downstream, having never been there, and be right when you arrive. Ask a noise-based generator the same question and it has to build eight thousand steps of world to answer.
The next buttons up top do exactly that arithmetic and jump you there.
The hierarchy is already in the sequence
Central-place theory says settlement sizes fall off in a power hierarchy, and
that is precisely the shape of a ruler sequence: half the columns qualify at
v₂ ≥ 1, a quarter at 2, an eighth at 3.
Nothing was tuned to make that happen. The spacing is
2k because the valuation says so, and the sizes stack in the
same order real ones do. Same argument as the rivers, one domain over.
Same bank without the buildings: The Riverbank → · the ladder itself: The Edge Run →
Buildings stand on the first flat ground back from the water, on whichever bank the hash
picks — so the law says where along the river and the ground says
exactly where. Everything else is bankScene, shared with
Sundial and The Riverbank.