Twenty-six words

A dense disk of particles dissolves into diagonal bands and stands again as a ghost of itself 192 steps later.

One world

The plates

Results

What this cannot establish

Twenty-six words

Twenty-six words, in this order: one, multiple, zero, set, point, pixel, vector, voxel, tensor, matrix, space, time, event, cause, effect, spacetime, reversibility, irreversibility, entropy, algorithm, input, output, stream, feedback, bit, reality bites back. The piece gives each one a form in a single reversible world made of bits, the lattice gas of world-first with walls and mirrors added, and builds each word from the ones before it.

Every word up to bit has a form inside the world: fixed by its law, by a way of looking at it, or by an intervention on it. Reality bites back needs something the world's model does not contain, a bit it did not see. Then the question is how much feedback it takes to get back in step:

Copying costs a stream. A model that runs the law and overwrites some of its bits with reality's each step never catches up below about 27% of the world per step, and always does above 28%: more than 4,400 of 16,384 bits a step, forever.
The law costs about 14 questions. A model that knows the law can wait while the unseen flipped bit spreads, then ask yes-or-no questions about single bits. After 48 steps it found the one bit among 16,384 in 13 to 15 questions in every one of 20 runs, mean 14.25, against log₂ 16,384 = 14.
The gas remembers its diagonals. This gas conserves 512 numbers on a 64 × 64 torus, among them the particle counts along every diagonal line, which travel one line per step. A dense disk dissolves, but its surplus survives as four travelling bands that keep crossing; ghosts of the disk return. The picture's entropy settles 359 bits below its maximum where the maximum-entropy state under all 512 invariants predicted 370 before the run, and swings by 369 bits every 32 steps, matching that prediction's phase profile with correlation 0.9999.
Space, two ways. One particle can reach, in t steps, a diamond |x| + |y| ≤ t. A push to the crowd spreads as a circle at 1/√2 per step, measured at 0.718 along the axes and 0.706 along the diagonals.
Order carries dimension. Counting a collision's ancestors by depth gives a tree near the event (growth 0.149 per step against 0.152 from theory) and a cone further back, whose growth gives a dimension near 3 in the square world (3.21–3.30) and near 2 in a strip four sites high (2.13–2.16).
The bit that was not kept. Run a drawing into smoke, turn every particle around and run back: it returns bit for bit. Get one bit wrong at the turn and the error equals that bit's forward effect over the same steps (ratios 0.99, 0.99 and 1.05).

Status, 29 September 2026. Of seven predictions written before any simulation existed, four held and three failed in part (P2, P3, P7); ten structural checks all pass. The interactive page has one plate per word, in the list's order, and computes everything in the browser.

One world

The HPP lattice gas (Hardy, de Pazzis and Pomeau, 1973) on a torus: four bits per site, one per direction. A step collides (a site holding exactly two head-on particles turns them 90°) and then streams (every particle moves one site). Walls reverse particles; mirrors / and \ turn them. Every step is a one-to-one map on the world's states, and turning every direction around makes the law run the history backward: Θ Fⁿ Θ = F⁻ⁿ. On an even torus the gas splits into two classes of sites, by the parity of x + y + t, whose particles never meet.

Working out the theory before any run showed that this gas conserves far more than a real gas does. Besides particle number and momentum it keeps the east-minus-west count of every row and the north-minus-south count of every column, separately on the two classes, and along every diagonal line x + y = ξ the number of particles moving toward larger ξ, which travels one line per step exactly. A collision E+W → N+S keeps one mover of each sign at its site, so nothing can change these counts. These extra invariants of the HPP model are known in the lattice-gas literature; the piece measures their consequences for its words.

The plates

#	Word	Form	What it shows
1	one	one particle in an empty world	it returns every 24 steps on a 24-wide world, and nothing happens to it: a collision needs two
2	multiple	k particles among 2,304 bits	C(2,304, k) possible worlds; collisions at 2ρ²(1 − ρ)² per site per step
3	zero	the empty world beside the full one	both stand still; the full world's holes move exactly like the empty world's particles
4	set	two particles collide, coloured by a convention	the world is the set of bits that are on; which particle went where is not defined
5	point	the world beside its 4,096 bits drawn as an image	one corner of a 4,096-dimensional cube; one step moves it about as far as an unrelated world
6	pixel	a camera counting particles per block	10^1,220 worlds behind one picture; two of them make different pictures one step later
7	vector	momentum per block, each row's momentum as a bar	256 momenta conserved; a mirror breaks them
8	voxel	a history as a block of voxels	erase all but one slice and the block grows back identical
9	tensor	P, P ⊗ P, P ⊗ P ⊗ P	the collision on three sites is the Kronecker product of the site matrix
10	matrix	the whole law of a 2 × 2 world	a 65,536 × 65,536 permutation; sorted by what it conserves it falls into blocks; every state returns within 12 steps
11	space	a pressure pulse, averaged over runs	a particle's diamond and the crowd's circle
12	time	the disk, its entropy over time	the bands keep crossing and ghosts of the disk return; whole small worlds return, larger ones effectively never
13	event	collisions against density	an event is where the law does something
14	cause	the bits whose flip changes a chosen site	at depth 6, about 8% of the light cone on average
15	effect	one flipped bit in twin worlds	the difference spreads at the light cone along diagonals, 0.78 along axes, and never vanishes
16	spacetime	the causal order of collisions	a tree near the event, a cone of dimension 3 or 2 further back
17	reversibility	draw, run into smoke, turn, run back	the drawing returns bit for bit
18	irreversibility	the same with one bit wrong at the turn	the error equals one bit's forward effect
19	entropy	the disk and a uniform gas	entropy settles where the world's invariants allow, not at its maximum
20	algorithm	two particles and two mirrors	one collision is a half adder; the mirrors are its program
21	input	the same with one input a step late	timing is part of the input
22	output	the adder run backward	the outputs determine the inputs
23	stream	typed text as particles moving along a row	a reader turns them back into letters
24	feedback	a loop of four mirrors	the line's end feeds its beginning: a delay-line memory of 64 bits
25	bit	flip one place on the loop	one flip changes one letter; nothing marks which flips were meant
26	reality bites back	a copying twin and an asking twin	copying needs over a quarter of the world per step; the law needs about log₂ H questions

Results

	Word	Registered	Measured
P1	space	ring speed along axes and diagonals in [0.64, 0.78], ratio in [0.90, 1.10], diagonal ring sharper	0.718 and 0.706, ratio 1.018; widths 11.0 (diagonal) and 12.5 (axis)	held
P2	effect	front speeds, interior plateau and a growth exponent in [1.8, 2.3]	speeds 0.783 and 0.704, plateau 0.202; exponent 2.32	failed, 1 of 4 parts
P3	spacetime	early growth in [0.12, 0.18] for square and strip; late dimension in [2.8, 3.8] and [1.7, 2.5]; the strip's peak earlier	0.149 and 0.116; 3.21–3.30 and 2.13–2.16; both peaks at depth 45	failed, 2 of 7 parts
P4	irreversibility	echo error / forward damage in [0.90, 1.10] at 16, 32 and 64 steps	0.990, 0.988, 1.048	held
P5	entropy	uniform deficit [160, 200] bits; disk [340, 430]; phase correlation above 0.9	183.0; 359.4; 0.9999	held
P6	reality bites back	copying threshold in [0.10, 0.30] on a grid of 0.05	0.30	held
P7	reality bites back	questions within ×1.5 of a formula at five waits; at most 21 after 48 steps	×0.80, 0.53, 0.56, 0.71, 0.93; 14.25	failed, 2 of 7 parts

The failures, at their size. P2: the damage count grew as t^2.32, 0.02 above the range; measured afterwards, its local exponent falls from 2.93 near t = 16 to 2.31 near t = 96, a transient on its way to 2. P3: in the square world the ancestor count matched the tree formula to within 3% out to depth 30, but a strip four sites high lets ancestors meet almost at once, since a particle moving across it wraps around in four steps: its early growth was 0.116, and its peak did not come earlier. P7: at waits of 16 and 24 steps the asking model needed about half the questions my formula predicted, because its best first question covers about four times the mean damage, not the mean. P6 held at the edge: a finer grid, run afterwards, puts the threshold between 0.26 and 0.28, about 12% below the mean-field 0.308.

Twenty-six words?

One world?

The plates?

Results?

What this cannot establish?

Twenty-six words

One world

The plates

Results

What this cannot establish