# The bench Instruments for playing with the tile. Local only, not deployed, not part of the publication moratorium: nothing here leaves the machine. Open `index.html` by double-clicking it. No server, no build step, no dependencies. Everything is a classic script so `file://` works. ``` kernel.js the tile engine — fold, strike, slots, grain, census, seams, birth folds test-kernel.js test vectors from the repo — run: node web/bench/test-kernel.js index.html instrument 1: the grain census main.js its views, controls, exports numberline.html the number line: one column per integer, strikes stacked numberline.js its views and controls dimensions.html instrument 3: the dimension of a hole dimensions.js its views and controls uniquebox.html instrument 4: the unique box (the stricter reading) uniquebox.js its views and controls ``` ## The dimension of a hole A number is k-dimensional when it can be built as a box of k factors each at least 2, which happens exactly when Ω(n) ≥ k. Primes are one-dimensional. This colours the tile's holes by dimension instead of by whether they are slots. See `research/OBSERVATIONS.md` entry 5 for the mapping to the standard names: the k-dimensional primes are the almost-primes P_{k−1}. Two facts the instrument is built around, both PROVEN in a line and both visible immediately in the picture: - Every hole of the tile has all its prime factors above p, so **the first hole of dimension k sits at exactly p_next^k**. Those are the ignition points, and the k = 2 one is the crystallization frontier: below it every hole is a prime. - The left chart counts only the tile's holes, the right one counts every integer over the same range. **The difference between the two is what the sieve did**, drawn rather than argued. Reach: up to T19 (width 9.7M, 1.66M holes, about 150 ms). T23 needs 223 million Ω values and is out of browser range. ## The unique box Instrument 4 is instrument 3 under the stricter definition (`OBSERVATIONS.md` entry 5a): n is a k-box number when exactly *one* unordered set of k factors multiplies to n, which is `{Ω(n) = k} ∪ {p^(k+1)}`. The only difference between the two readings is that thin exceptional family, so this instrument makes it the subject. Prime-power holes box uniquely at both k and k−1, because identical factors leave no choice about grouping. They are drawn as a white rim on their band, marked in the strip, counted per class in the table, and listed individually in their own panel. Measured at T13: 39 of 5,760 holes, 0.677%, all of them q^e for a prime q above 13 (34 squares and 5 cubes). The disagreement between the two definitions is real but vanishing. ## The number line One column per integer. Every stacked prime strikes its multiples and adds a block, so a column's height is how many stacked primes divide that number, and height zero is a hole. Two holes two apart get a bar underneath, coloured by the fold that gave birth to that slot. The same colour always means the same prime, whether it is striking or giving birth. A slot's bar is filled below the frontier p², where a hole is a certified prime and the slot is a real twin prime pair, and hollow past it. Worth doing once: stack 2, 3, 5, 7, 11, 13, then press "tile edge p#" and step along with the buttons. The slots born at fold 11 appear only near multiples of 210, which is the Seam Lemma made visible: the natal set is born at the seam. ## The kernel One engine, plain arrays, no rendering assumptions. Everything else is a view of it, and the scroll page's exhibits will be too. `buildSlots(levelIndex, d)` folds from the 2-tile up, one prime at a time: lay p copies end to end, strike the residues that would put r or r+d on a multiple of p, keep the rest. That is the Copying Theorem run as an algorithm, so the slot list is generated the way the framework says the family grows, rather than by sieving positions and checking them afterwards. Reach: direct hole rendering to T19 (width 9,699,690), slot lists to T23 (7,952,175 slots, about half a second end to end). T29 needs 215 million slots and is out of browser range. ## Verification `node web/bench/test-kernel.js` checks the engine against numbers already committed elsewhere in the repo, not against itself: - widths and censuses, including T31's 6,226,553,025 - the three houses: the slots of T5 are exactly 11, 17, 29 - T7's grain, verbatim against GLOSSARY.md - T11's grain census, verbatim against GLOSSARY.md - the G2 ladder T2..T29 against the twin Jacobsthal list - the d ↦ G_d split: G8 = 198 against G2 = 150 at 19#, from README.md - every T13 slot sits on a hole with its partner on a hole - gaps sum to the width at every level tested The page reruns three of these in the browser on load and says so in the header. It also checks, live at whatever level is on screen, the two laws proved in `research/grain-census.js`: count(6) = ∏(q−4), the forced ratio 8·count(6) = 3·count(12), and the word rule that "6,6" never occurs. ## What instrument 1 turned up Nothing new, which is worth recording plainly. Playing with the per-fold multiplier column reproduces, from the folding side, exactly what `research/grain-census.js` had already derived from the CRT side on 2026-08-14: gap sizes 6 and 12 multiply by exactly p−4 at every fold from p=11 up, with the p=7 exception explained by 14 ≡ 0 mod 7 collapsing four forbidden residues to three; larger sizes exceed p−4 because folds also fuse smaller gaps, which 6 and 12 are immune to since the minimum gap is 6 and two 6s are never adjacent. Two things came out of that anyway. The agreement is a real cross-check: two independent implementations, one folding and one inclusion-exclusion, produce the same censuses digit for digit. And GLOSSARY.md still said the grain census "awaits a law", which was true when written and stale by the time of the grain session; that line now points at the file that settled it. The consequence for the bench: instrument 1's research target is closed, so **instrument 2 (where the monster gaps live) is the top of the queue.** The groundwork is already here — `topGaps`, the seam overlay, and the deviation view that shows positional structure once a pixel holds more than eight slots. ## Notes on the views **The tile.** Below about eight slots per pixel this plots density. Above it, density saturates to a solid block and hides everything, so it switches to deviation from the mean and says so. The deviation view is the one worth staring at: at T19 the whole tile fluctuates inside ±3.4%. **The grain census.** Ghost bars are the previous level's shares, so a fold's effect on the distribution is visible in one picture. Green bars are gap sizes that did not exist one level down. **The table.** The `×` column is count at this level divided by count one level down. The reference value is the fold multiplier p−2 shown in the readouts. **Export.** Every picture dumps as numbers, to the textarea and the clipboard. Every knob is in the URL, so an interesting configuration is a link.