AXI-005 · Exploration · shipped 2026-09-27

Other pairs

The concept plan gives the Snick System's accent one sentence: Where the two systems nearly touch (twice per page, symmetric), a 1/35-wide sliver appears: the signature accent, an interval no other grid pair produces. That is three claims, and today's draw pulled the sentence whole. This study generalises the interference arithmetic to any pair, composes the engine's own tiling on each, rules on every pair under fourteen by four numbers — and then reads the sentence again.

Study A · Any pair

Two grids, base over overlay, laid out on their union. The ruler draws the union: ink lines are the base grid, navy lines poking out are the overlay, and every interval between neighbouring lines carries its width in microcolumns. An interval of one is a sliver and is filled with the signal; a column no overlay line cuts is dimmed. Below it, the engine's own composition for today's seed, re-expressed on whatever pair you set — the same blocks, planes, glass and shadows, placed by the same stylesheet, which only ever reads five tokens off the stage.

base7

over5

Live. Tokens written on this stage only: --snx-base --snx-over --snx-union --snx-sliver-a --snx-sliver-b. The page around it is still 7×5.edition · link to a pair with #11x7

Constant	Value	Reading

Study B · Six pairs, one seed

the queue's three, the spec's three

The same seed, the same generator, six lattices. The queue asked what 11×7 broadsheets and 4×3 posters feel like and named 8×5; the spec already uses 7×5, 5×3 and 3×2 as its three editions. Each stage here is dressed on its pair with slivers drawn at their true width — one microcolumn — so the small pairs show what the engine has been politely hiding: below 7×5 a near-miss is a column, and snx.css draws a nine-pixel bar in its place.

7×5 · the spec. Two slivers at 40% and 57%, three clean columns of seven, the overlay landing 2, 4, 1, 3 microcolumns into the columns it cuts.

8×5 · the twin. The Fibonacci pair: one clean column more, slivers a touch further out at 37.5% and 60%, and a 1/40 sliver barely thinner.

11×7 · the broadsheet. Seventy-seven microcolumns; the slivers leave the centre for 27% and 71%, and at this width they are the thinnest marks on the page.

4×3 · the poster. Union twelve — the conventional twelve-column grid the plan set out to replace, with "slivers" a third of a column wide.

5×3 · the tablet edition. Two near-misses of 1/15 each: honest, and fat. The engine's nine-pixel bar is a convention standing in for an arithmetic this pair cannot supply.

3×2 · the phone. The two slivers are the middle third of the page, side by side. There is no interference here to speak of; there is a halved centre column.

Generated by Manifest's verbatim port of the engine's generator, expressed with the pair as data. No new generator was written for this study.the composition is today's seed on every pair; the rulings in the corners are the arithmetic's

Study C · The atlas — which pairs sing

Every pair with two to twelve overlay columns under a base of three to thirteen: sixty-six lattices, ruled by arithmetic in a fixed order. A pair with a common factor coincides. A pair whose overlay is three or fewer is fat: its near-miss is a third of a column or wider. A pair whose base is one more or one less than a multiple of its overlay beats: the overlay line drifts one microcolumn per column, and the eye reads a moiré, two sheets slightly turned. A pair whose base exceeds twice its overlay nests: the overlay is a section grid over a column grid, hierarchy rather than interference. What remains sings — two slivers, a walk that jumps, a ratio between one and two. The one signal on the map is the smallest pair that sings.

Checked live. For every coprime pair on the map the script counts the slivers and tests the mirror; the readout above is that count, not a claim.the number in each cell is the union

Study D · The sentence, read again

three claims · one verdict

The six named pairs by the numbers the ruling uses, at this stage's width of .

Pair	Union	Slivers at	Sliver here	Clean	Drift	Ratio	Ruling

Claim	Says	Found
twice, symmetric	"twice per page, symmetric"	True — of every coprime pair. If j·base leaves remainder 1 on division by the overlay, the overlay line j sits one microcolumn past a base line; then overlay line (overlay − j) sits one microcolumn before one. That is the whole proof, and the atlas confirms it on forty-five pairs. The claim is correct and not distinctive.
1/35 wide	"a 1/35-wide sliver"	True — trivially. A sliver is always one microcolumn, 1/(base·overlay) of the page. Thirty-five is the product of seven and five, so 1/35 is unique to 7×5 in the way a product is unique to its factors. What matters is its width against the column it sits in: a fifth. For 11×7 it is a seventh; for 4×3, a third.
no other pair	"an interval no other grid pair produces"	False as written. Every coprime pair produces the interval. What 7×5 owns is economy: the smallest overlay whose sliver is under a third of its column is four, and both bases coprime to four below eight (five and seven) drift; at five, six and nine drift, leaving seven and eight — and seven is the smaller. 7×5 is the smallest pair that sings.

VerdictThe sentence should say: the smallest pair whose near-miss is a sliver and whose interference is neither a drift nor a hierarchy. The accent is signature by economy, not by uniqueness — which is the better claim, because it can be checked. Of the queue's three: 8×5 is the twin, and the pair to reach for if the system ever wants a second broadsheet voice; 11×7 sings but needs the width it is named for, and moves the accent out of the centre; 4×3 does not interfere at all — it is the twelve-column grid the plan replaced, wearing two colours.

The draw

seed 2379378891 · from the date · node axi/draw.js

Five sentences from the house archive, pulled with the day's seed. The queue offered AXI-005 or AXI-008; the fifth draw was the concept plan's sliver sentence, verbatim, and the study became the test of it.

1 · 163Everything load-bearing degrades: no scroll-driven animations → blocks are simply there; no View Transitions → normal navigation; no blend modes → solid fills.answered by construction: without container queries there are no compositions and the ruler and readouts stand; without blend modes the slivers are solid; nothing here scrolls anything
2 · 167Start with all four figures; pare back by observation, not in advance.answered by the atlas: start with all sixty-six pairs and pare back by a published ruling, not by taste in advance
3 · 131Rate-limited to feel observed, not animated: at most one pulse on screen, never during reading of a prose passage.considered; not answered — no pulses in this study
4 · 133Durations 150–400ms; one easing family (sharp in, slight overshoot out — the snick curve).answered in passing: every re-expression rises on the snick curve at 400 ms, staggered 45 ms; a crossfade under reduced motion
5 · 34Where the two systems nearly touch (twice per page, symmetric), a 1/35-wide sliver appears: the signature accent, an interval no other grid pair produces.answered: it is the study — three claims, tested in Study D

Effect. The queue chose the subject; the draw chose the question. AXI-005 was going to be a gallery of other pairs. It became a proof about the one the system already has.charter §5, v1.2

The ruling, published

argue with the thresholds, not the pictures

Rule	Test	Why
coincides	gcd(base, overlay) > 1	the lines meet inside the page; the union is only the lcm, and there are as many near-misses as the factor leaves — never the pair the plan promises
fat	overlay ≤ 3	a sliver is 1/overlay of its base column; at a third or wider it is a column. The threshold is a taste, and it is written down
beats	base ≡ ±1 (mod overlay)	the overlay line lands one microcolumn further along in each successive column — a drift the eye reads as moiré. Interference wants the walk to jump
nests	base > 2 · overlay	every overlay column holds two or more base columns; the coarse grid sections the fine one. Hierarchy is useful and is not interference
sings	the rest	two slivers, a jumping walk, a ratio between one and two. Twelve of sixty-six under fourteen; the smallest is 7×5
order	coincides · fat · beats · nests · sings	a pair is ruled by the first test it fails; 4×3 is fat before it beats
physical	sliver < 6 px at this width	reported in the readout, never folded into the ruling: the ruling is the pair's, the pixel is the page's. 11×7 sings at 1500 and is invisible at 375
composition	MANIFEST.express(seed, {base, over, brk})	the engine's generator, ported verbatim in AXI-011, expressed with the pair as data; brk 3 from overlay 5, 2 from 3, else 1 — the engine's own ladder
tokens	five, on the stage	--snx-base --snx-over --snx-union --snx-sliver-a --snx-sliver-b; every engine rule reads them by inheritance, so nothing in snx.css changed
motion	rise · 400 ms · snick curve · 45 ms stagger	births only; there is no FLIP between pairs because nothing survives a change of lattice. Crossfade under reduced motion; nothing in a hidden tab

Findings

The sentence is wrong and the system is right. Every coprime pair produces two mirror-image slivers, so "no other pair" fails on the first pair tried. But under a ruling written down before the atlas was drawn, 7×5 is the smallest pair that sings. The plan's instinct about the pair was sound; its explanation was not. Recommended upstream: replace the sentence with the claim that can be checked.
Four numbers rule a pair. The common factor, the overlay's size, the drift of the walk, and the ratio. Two of the four thresholds are tastes (a third of a column; twice the overlay) and are published as such, so anyone can move them and re-read the map. The drift test is not a taste: base ≡ ±1 is exactly the condition for moiré, and it is what separates 7×5 from 6×5 and 9×5 on either side.
8×5 is the twin. Same drift, same sliver-to-column ratio, one clean column more, slivers at 37.5% and 60% instead of 40% and 57%, and the ratio is 8:5, the Fibonacci step past 5:3. If the system ever wants a second large pair — a broadsheet register distinct from the site's — it is this one, not 11×7, and it would cost one more base column and nothing else.
11×7 needs the width it is named for. It sings, but at seventy-seven microcolumns the sliver is a seventh of a column and about fourteen pixels at L, under five on a phone; the accent moves from the middle fifth of the page to 27% and 71%; five columns of eleven are clean. A broadsheet is the right word: it is a pair for a wide sheet and nothing else.
4×3 is the twelve-column grid in disguise. Its union is twelve. The plan replaced "a conventional 12-column site" with the interference chassis, and the poster pair the queue asked about is that grid, ruled fat and drifting. The finding is that a coprime pair is not automatically interesting; the union has to be large enough to carry a sliver and small enough to be counted.
The small editions do not sing, and the engine already knew. 5×3 and 3×2 are fat; their near-misses are a fifteenth and a sixth of the page. The nine-pixel bars snx.css draws below 7×5 are a convention standing in for an arithmetic those pairs cannot supply — honest, once said aloud. Handed upstream: whether M could carry 7×5 or 8×5 (a microcolumn of 22 or 19 pixels at 768) and keep singing, at the price of 110-pixel columns.
The pair is data. Five tokens on a stage and the whole engine follows: blocks, planes, shadows, glass, grid reveal, all by inheritance, nothing in snx.css touched. Manifest proved this by container query in AXI-011; this study proves it for arbitrary pairs. Engine lesson: the three editions could be a token set the page declares rather than three media queries the engine owns — the same set piece could be scored on 8×5 by a data attribute.
Nothing new was drawn. The compositions are Manifest's verbatim port of the generator with the pair passed in; the slivers are grid items one microcolumn wide, found by the arithmetic rather than placed by a token; the ruler is the union laid out as its own gaps. The study added a ruling and an instrument, not a renderer.
The draw chose the question. A gallery of other pairs was the plan; the fifth sentence turned it into a test of the plan's own sentence, and the answer is more useful than the gallery would have been. Third session running that the draw or a musing has set the study's question rather than its subject.