AXI-016 · Exploration · shipped 2026-10-04

The torus

A torus is a manifest whose left edge is its right edge. Thirty-five is a multiple of seven and of five, so the two grids coincide at every face boundary and a seam between faces is a line of both grids — invisible as a seam. This study makes the drum Dan described: four faces of the 7×5 pair on one circle, generated once by the engine's own generator and rolled by the reader's scroll with no end; then the point of coincidence rolled to the centre of the window, where the generator has never put it; then the drum set at plane −1, behind a window cut in the sheet, rolled by the page — the plan's one sanctioned parallax. From a musing by Dan (2026-09-30); the draw supplied the third study.

Study A · The drum

One torus, page-wide, on today's seed. Its lattice is four faces of 7×5 on a circle of 140 microcolumns, tiled by the engine's BSP in one pass over twenty-eight base columns and then turned by a random number of columns, so that the wrap is crossed by a tile like any other seam. Scroll it sideways — trackpad, shift-wheel, the scrollbar, or the arrow keys when it has focus — and it never ends: when the roll settles in a cloned face it jumps by four faces, and because the pixels are identical you cannot see it do so. The roll snaps lightly to two kinds of stop: a face-aligned roll, where a seam sits on the window's edge, and a seam-centred one. The corner reads the roll; seams draws every coincidence of the two grids; grids reveals both.

roll

seam

show

One signal per window. The engine's law is one signal moment per viewport; on a circle it becomes: no window one face wide, at any roll, may hold two. Two signals need two gaps of a face each, three need three, four are impossible. The drum keeps the engine's one and promotes the smallest tiles the law allows; the readout counts the signals in view as you roll.edition · pair by the torus's width · rows by the page

Study B · The seam at centre

The same torus, seed 16, in two fixed windows. The left is rolled to zero: a seam on each edge, which is every composition the generator has ever drawn — its stage edges are the coincidence of the grids. The right is rolled by half a face, seventeen and a half microcolumns: the seam stands at the centre of the window and the twin slivers, which the plan put near centre stage at 14 and 20, have gone to the edges. The note beneath is the instrument: the census of the signal law at every half-microcolumn roll, the tiles that cross each seam, and the lines of both grids in each window.

Roll 0. The generator's view: both edges are seams, and the slivers flank the centre (14 and 20 at 7×5).

Roll half a face. The seam at centre; both grids mirror about it; the slivers are the outermost pair (2.5 and 31.5 at 7×5); no grid line on either edge.

measuring…

Study C · Through the window

The plan's third plane has had one tenant on this site — the dusk rectangle the engine draws in a window — and one rule: recessed texture, seen through windows cut in plane 0, with inset shade from the cut edge, moving slower than scroll, "the one sanctioned parallax." The draw pulled that sentence. So the drum goes behind the sheet: a window four columns wide in an ordinary set piece, the torus inside it at plane −1, muted, its signals demoted to sky because the signal moment belongs to the sheet, and its roll driven by the page's own scroll on a view() timeline. As the window climbs the viewport the drum turns half a face, so it enters with the seam at the edge and leaves with the seam at centre. The readout gives the ratio.

plane −1What is behind the sheet moves slower than the sheet.

Slower than scroll. The window travels one viewport height plus its own; the drum travels half a face — 317px against 1,124 at L, a ratio of 0.28; 0.27 at XL, 0.17 at M, 0.15 at S. Under prefers-reduced-motion the drum does not roll and rests with the seam at centre.edition · timeline —

Interstitial · comfortable measure

What the roll changes

The generator has always drawn with the seam at the edge. Its stage is one face; its left and right edges are where the base and overlay lines agree; and everything it knows about the interference pattern — the slivers at 14 and 20, the fat centre, the four near-coincidences between — it knows relative to those edges. Close the stage into a circle and the edge becomes a choice. At roll zero nothing has changed. At any other roll the window shows a stretch of the pattern the generator has never framed, and at half a face it shows the one stretch with a mirror axis down the middle: base lines at 2.5, 7.5, 12.5 and overlay lines at 3.5, 10.5, 17.5, reflected, with the slivers as the outermost pair and no line of either grid on the window's edge.

Dan's intuition was that this would open new geometries, and it does, but not by adding anything. The torus has the same tiles, the same two grids and the same rule for colour as the stage; what it adds is a parameter the stage did not have — where the frame falls on the pattern — and a kind of block the stage could not have, one that crosses a seam. The drum above has several. A tile that runs from one face into the next is still a rectangle on both grids; it is simply a rectangle the generator's bounded stage could not contain. The unbounded manifest Dan asked for turns out to be the bounded one with its two ends identified, and a roll.

The draw

seed 1618814878 · from the date · pool 191

Dan gave the session a free run. The queue held six studies and the draw was the only other voice; five sentences, seeded by the date:

1 · 178Every composition below is generated from a seed and snaps to the union of both grids.answered: the drum is one composition from one seed on a union of 140 columns, and every piece — including the cut halves of a tile that crosses the wrap — snaps to it
2 · 174P5 Motion & dashboard layer; polish; launch 1–2 sessionsconsidered: the plan's fifth phase; the dashboard layer here is the readout, and the motion is the reader's own
3 · 91XL ≥1440 7 × 5 35 14–15 · 20–21 Full interference; generous set piecesanswered: the sliver columns named in the plan's edition table are blocks on the strip, two per face, and roll with it; a tile across a seam is as generous as a set piece gets
4 · 153The prose-passage rhythm (§1, principle 04), the one-signal rule, and a density budget per viewport are the counterweights.answered as law 3: on a circle the one-signal rule becomes "no window at any roll holds two," which caps a drum at three signals; the note in Study B is the proof
5 · 69−1 Recessed texture: muted imagery, ambient color, background data. Seen through "windows" cut in plane 0 (clip regions); inset shade from the cut edge; moves slower than scroll — the one sanctioned parallaxanswered as Study C and law 7: the drum behind a window, muted, without a signal, rolled by the page's scroll at about half its speed

Effect. The queue said torus or any of five others; the draw's fifth sentence said plane −1 and parallax, and the torus is the one queued study that is rolled by scroll. It chose the study and then wrote its third part. First draw since AXI-011 to tilt rather than furnish.charter §5 · draw.js

The rules, published

argue with the numbers, not the pictures

Constant	Value	Why
circle	F × union microcolumns · 140 at 7×5, four faces	the two grids coincide at every multiple of the union, so every face boundary is a line of both grids and no seam is special
pair	7×5 ≥ 540px · 5×3 ≥ 270px · 3×2 below	Manifest law 2, by the torus's own width; the rows are the page's rows, because the vertical is tame
tiling	the engine's BSP, once, over F × base columns · then turned by ρ base columns	a verbatim port; the turn (ρ from the same rng, after the tiling) makes the wrap a seam like the others — crossed by whatever tile happens to straddle it
adjacency	on the circle	the engine's no-like-neighbours rule compares a tile with its neighbours' images one circle away as well
breakouts	1–3 per face, placed anywhere on the circle	the engine's loop run F times; a breakout may straddle a seam, the wrap included; the ledger's objections are checked against tiles on the circle
signals	≤ 1 per window at any roll · ≤ 3 per circle	two signal tiles need both gaps between them ≥ one face; the engine's one is kept and the smallest tiles the law allows are promoted; the census checks all 280 half-microcolumn rolls
roll	--roll, microcolumns, continuous	the only moving part; the stylesheet translates the strip by −(union + roll) microcolumns
strip	F + 2 faces · the last face cloned before the first, the first after the last	every block is drawn at each of its images and cut at the strip's ends, so any window of one face sees a whole circle's worth of content
jump	± F faces when the scroll settles in a clone	the pixels are identical before and after, so the jump cannot be seen; a scroller clamped at either end jumps at once
stops	every half face · proximity	face-aligned and seam-centred rolls both snap; a free roll is still allowed
units	half microcolumns	the seam-centred roll is 17.5 columns; the strip's grid is scored in halves so that stop exists as a line
slivers	blocks at 14 and 20 of every face · z1 · multiply .85	law 6: structure rolls with the content; full columns at 7×5, 5px bars below it, as the manifest does
plane −1	saturate .78 · brightness .95 · signals → sky · roll 0 → union/2 over entry to exit	the plan's recessed plane: muted, seen through a cut, slower than scroll; no signal below the sheet
reduced motion	no roll behind the window · instant jumps · smooth scroll off	the reader's own scroll is never motion the page added; the page's parallax is

Findings

The seam can vanish because 35 is a multiple of both grids, and the wrap too. With the tiling turned after generation, the wrap is crossed by a tile like any other seam — seed 16 at 7×5 is turned by three columns and has two pieces across the wrap and one across each of the other seams — and the drum's four seams are indistinguishable by eye until the seams switch draws them. Roll the drum through a whole turn and the only evidence of a wrap is the readout returning to zero.
The roll is a parameter the stage never had. The generator frames the pattern with a seam on each edge, always. At any other roll the window shows a stretch it has never framed; at half a face the stretch is mirror-symmetric about the centre line, with the slivers as the outermost pair at 2.5 and 31.5 and no line of either grid on an edge. Dan's "new geometries" are not new tiles but a new frame — and one new kind of block, the tile that crosses a seam.
One signal per viewport caps a drum at three. On a circle the law becomes "no window one face wide, at any roll, holds two signal tiles," which requires a gap of one face between every pair of signals; four signals would leave no room for any tile. The census in Study B checks every half-microcolumn roll: seed 16 carries two signals at 7×5 and never shows two in a window, and 122 of its 280 rolls (44%) show none; the day's drum carried three on 2026-10-04 with 43 of 280 rolls empty. A drum is between signal moments for most of a turn, which is the law's price on a circle and, read the other way, what makes the moment one.
The engine's signal breakouts are moments too, and the law vetoes them. The generator may paint a plane +1 breakout in the signal colour — the second-signal fault the sketchbook found in two mornings of three. On the torus an opaque signal breakout counts as a moment: it is kept if it has a face's gap from every other, and otherwise goes navy, as the sketchbook's veto would have it. The day's drum had one vetoed.
Plane −1 carries no signal. With the drum behind a window the sheet's own signal block and the drum's signals would share a viewport two rolls in three. The rule adopted — the recessed plane demotes its signals to sky — keeps the law without touching the sheet, and makes the mute the plan asked for do structural work as well as tonal.
The one sanctioned parallax can run sideways. A view() timeline on the strip, one @property for the roll and one keyframe pair are the whole mechanism; the drum turns half a face while the window crosses the viewport: 317px of roll against 1,124px of scroll at L (0.28), 0.27 at XL, 0.17 at M and 0.15 at S. The plan's clause "moves slower than scroll" was written for vertical drift; a horizontal roll satisfies it with the same economy and does not fight the reader's scroll direction, which vertical parallax always slightly does.
Half microcolumns are a unit. The seam-centred roll is a half-column roll, so the strip is scored in halves: 420 columns for six faces at 7×5, 180 at 5×3 — and "half a face" has to be said in the score as a word, not a number, because it is 17.5 at one pair and 7.5 at the next. The memory of the sky study's half-row baseline unit suggests the same for columns — a --snx-unit in both axes — and this is the second study to need it.
The engine's depth budget is per face. Run as written over a circle four faces wide, the BSP stopped at the same depth and drew eight tiles across 140 columns — tiles two faces long. The circle now starts log₂(F) levels deeper, which is the one declared change to the port: seed 16 draws 36 tiles at 7×5 and 29 at 5×3, about nine a face, which is the engine's own density.
Feed/score holds, with the manifest's exception. The script writes a start, a span and two rows per piece and ten data attributes per torus; the pair, the translate, the mute, the seams, the gridlines, the snap stops and the parallax are all stylesheet. The jump in the scroller is the script's, as the manifest's FLIP was, because a scroll position is state the stylesheet cannot hold.