◈ THE SEAM RANGES ◈
Lars Rocha's address space has two mirrors in it. One folds on a lattice of powers of two; the other folds on the squares. Each one on its own cuts often. Put them out of true and ask where they still agree, and almost everywhere the answer is nowhere — but not quite nowhere. The places where both cut are the seams, and a seam's position is settled by arithmetic alone: no table, no seed, no spawn rate, and the same seam in the same place for everyone who walks there.
The line mirror cuts at j when 2^w divides
j+1. The diagonal mirror cuts when j+1 is a perfect square. Both, then, exactly when
j+1 = s² and 2^w | s² — which needs v₂(s) ≥ ⌈w/2⌉. So:
two mirrors out of true agree only at
s², for s a multiple of
2^⌈w/2⌉.
width w | seams in 65,536 | one in | first four seams at |
|---|---|---|---|
| 4 | 63 | 1,040 | g = 15, 63, 143, 255 |
| 6 | 31 | 2,113 | g = 63, 255, 575, 1023 |
| 8 | 15 | 4,369 | g = 255, 1023, 2303, 4095 |
| 10 | 7 | 9,362 | g = 1023, 4095, 9215, 16383 |
| 12 | 3 | 21,845 | g = 4095, 16383, 36863, 65535 |
The detuning is the only knob, and it moves on a clean ladder. Widths pair up — 7 and 8 give
the same, 9 and 10 give the same — because only ⌈w/2⌉ counts. Turning a mirror one notch further out of
true changes nothing at all; two notches halves the seams.
And the gaps grow. Consecutive squares spread, so each walk between seams is longer than the
last by exactly 2·step² — 32 cells at width 4, 128 at width 6, 512 at width 8. Never varying, and
knowable before you set off. That single fact is what both games below are built on.
Hills to climb, and on the highest of them the seams. They start at g = 0 deliberately:
the beat's local density falls as 1/√g, so out at two and a half million the gap is 25,920 cells and
nobody would ever see two.
g=15, gaps of 48, 80, 112, 144. Close enough to walk one to the next and watch the spacing grow. The best range for seeing what the law does.
◈ The Seam RangeWidth 6, thirty-one seams, the first at g=63, gaps growing by 128 each time. Hills, rivers, prizes on the summits and a warp gate on the biggest.
◇ The Wide SeamsWidth 8, fifteen seams, the first at g=255, every gap 512 cells longer than the last. Fewer marks, longer country, every one provable before you arrive.
In any of them, Next seam ◈ on the travel bar flies you to the next one — jumping mark to mark by the law rather than searching the ground between, which is why a range costs nothing to enter however far out it runs.
The marks are far apart and provable in advance, so the useful virtues are reach, climb, and knowing exactly how far it is. Each reads the next seam from a different distance.
A stretch of ground, and one question: does a seam fall inside it? The law settles this exactly, so there is a right answer and no need to go and look. Guessing scores about half. Working it out scores all of it.
Press New round to begin.
Reach n seams in order. Because each leg is longer than the last by exactly
2·step², the whole route can be worked out before you move — which is the point. The planner below
gives you the route; the running is done in the range.
The placement law was not invented for this. It was already in
cards/mirrors.md as a result about two folds on an address. What happened is that a module written
independently in the spindexin workspace — landmarks.cjs, "exact shared-seam indexing" — turned out to
select the identical point set, checked at widths 4, 6, 8, 10 and 12: 63, 31, 15, 7 and 3 agreements in
65,536, the same points and not merely the same counts.
So a seam's place is settled before anything is drawn, and that is the whole claim. No claim is made about how it looks, or about what anyone sees in it. The arithmetic says where; the rest is a range to walk.