The Prime Rails from the air, region 0 of column 2,600,000. Every odd prime hangs off its octave's Mersenne ceiling (3 mod 4, the L rail) or Fermat floor (1 mod 4, the R rail) by a wrap that is a multiple of 4; the two rails run on their own lanes with a station pad at every prime, a bridge where a prime has a partner (front XOR 5 is the twin p−2 or the sexy p+6), a Cunningham tower where 2c+1 keeps being prime, and the gap valley opens beside a rail only where a gap opens, with a plain between. A new world in Wander; its law in sitoolz.primerails, 16,130 values identical between Python and the browser.
Standing on the L rail at lane −24. A station pad at every prime of the rail; where a prime has a partner (front XOR 5 is the twin p−2 or the sexy p+6) a bridge crosses to the other rail or runs along this one; where 2c+1 keeps being prime a Cunningham tower stands as tall as the chain (89's ladder is six rungs). Proth ridges cross all lanes where k is rich.
The Fold Marches at dawn. The XOR laws as ground: relief from mask agreement between neighbouring primes ((q^m) − (p^m) = q − p exactly when p and q agree on every bit of the mask; mask 5 here, 1 is flat, 42 is badlands), the two-squares realm north of the road where primes 1 mod 4 raise domes and the Gaussian realm south where primes 3 mod 4 raise spires, and vault doors sunk where the square remainder folds to a nested square.
On foot in the Fold Marches. The ^5 sorter sends the two even lanes to opposite sides of the two-squares theorem, so the road between the realms is the sorter drawn as a line; the fork behind the domes and spires is the mod-4 fork: a composite peels as a² − b², a prime 1 mod 4 is a² + b², a prime 3 mod 4 is neither.
The Vortex Sea. An exact integer fluid, frozen: flow2z's two-dimensional vorticity on an integer stream function, divergence exactly zero and the total vorticity conserved to the last bit, run for 48 steps from one seed and read as ground. The whirls are hills, the channels are sea, tiled every 256 cells, the same on every machine (25,142 values in the fixture, including the dyadic shell cascade to the last bit).
On the water of the Vortex Sea, a jetski world. The stream function's zero line is the shore; the prizes stand on the whirls by the prize law and the gate on the biggest hill, as in every ruler-land range.
The Vortex Sea as the showcase draws it: 512 columns by 192 lanes through terrain.window, colour by class, drawn twice (Python and JavaScript) and compared pixel for pixel: identical.
The Prime Rails as a map: the gap valley from above. The stripes are the gaps between consecutive primes near 2,600,959; a wide grey band is a wide gap, a green band a run of close primes. Drawn twice, identical.
Three postcards in Lars's Hamlet code: two warp addresses (rulerhills-1jqwv-mgzs-j and one more) and a Light Cone 2 link, encoded by sitoolz.hamlet and its JavaScript twin. Anchors where v2(x+1) ≥ 4 and v2(y+1) ≥ 4, layers replicated with period 2^(k+1) so a torn scrap decodes its layer without its position, Reed–Solomon over the rest: 11,009 checks identical across the twins.
The Square Vault, a demo game from the folds engine: every tile is a perfect square scrambled by one hidden XOR mask; flip the mask's bits until every tile is a square and the door opens, then peel a nested square root by root down to the ground seed (0 or 2). Lars's Square Lock and nested squares, dealt by the indexed generator so the same name opens the same vault everywhere.
The walkable world under Lars's Celestial Zones shader: a domain-warped nebula, a breathing hex lattice masked by the nebula's density, oracle pulses and three parallax star layers, turned by the camera; by day it fades to Wander's blue. Seven layers from one integer, and now a sky.
The Lab on the site: the unblackboxing reports rendered a tab each. Given 20,000 columns and no law, WSRF found the prize rule as rank ≥ 4.29 and sqrem_even, the settlement rule as rank ≥ 6.05, and walkable as height ≥ 21.5 with the water line at 22: accuracy 1.000 on all three, the laws in the forest's own variables. Every number comes from a script in sitoolz/experiments that reruns in under two minutes.
The Ruler Land page with the three new ranges at the end of its seventeen: each card draws the law's own profile across the gate of region 0, names the knobs (land primerails · unit 2 · cap 24 · lane 24 · tower 6; land folds · mask 5; land vortex · L 6 · steps 48), counts the prizes and names the gate, and offers a Fly link, the gate's warp and a walk in.
The Octave Valleys at dawn, region 0 of column 522,240, and the beacon pillar at 524,287 = 2^19 − 1, a Mersenne prime. One valley per octave of the number line, mirrored: the north wall rises with the floor wrap of the nearest prime (read up from 2^k + 1) and the south wall falls with its ceiling wrap (read down from 2^(k+1) − 1), and the two always sum to 2^k − 2; past the beacon the valley flips into the next octave. The plain is dithered by the rail cycle's dropout mask. sitoolz.octaves: 24,065 values identical between the twins, the beacons proven to 2^127 − 1 by Lucas–Lehmer.