Predictions, written before any simulation

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

The subject

The world

What the gas conserves

Theory

Structural checks

Predictions

The 26 plates

What this cannot establish

World

The HPP lattice gas (Hardy, de Pazzis and Pomeau, 1973), the gas of world-first, on a W × H torus: four bits per site, one per direction (E, N, W, S). A step collides (a site holding exactly E+W becomes N+S and back) and then streams (every particle moves one site).

Additions

Walls (a particle about to enter one reverses in place) and mirrors: / swaps E↔N and W↔S, \ swaps E↔S and W↔N, at the mirror's site instead of the collision. Every step stays a bijection.

Reversal

Θ = (reverse every direction) ∘ (the site's own permutation). Θ Fⁿ Θ = F⁻ⁿ for tori, walls and mirrors.

Density

ρ = 1/3 per bit unless stated: the density at which a particle collides most often, ρ(1 − ρ)² = 4/27 per step.

Classes

On an even torus a particle's class, (x + y + t) mod 2, never changes, and particles of different classes never meet. The gas is two gases that pass through each other.

Invariant

Why

x momentum (E − W) of every row, per class; y momentum (N − S) of every column, per class

a collision turns one E and one W of one class in one row into N+S, so E − W in that row is unchanged

counts along diagonal lines, travelling

with ξ = x + y, the movers E and S advance ξ by one per step and W and N retreat it by one; a collision E+W → N+S keeps one of each at its site. So the number of E or S movers on line ξ at time t equals the number on line ξ − t at time 0, exactly. The same holds for η = x − y with E and N.

Quantity

Value

From

collision events per site per step

2ρ²(1 − ρ)² = 0.0988

product measure

collisions per particle per step

ρ(1 − ρ)² = 0.148

sound speed

1/√2 = 0.707 in every direction (pressure = n/2)

linear hydrodynamics

light cone

the diamond |dx| + |dy| ≤ t: 1 site per step along the axes, 1/√2 along the diagonals

one site per step along an axis

damage from one flipped bit: branching

a differing bit meets a site where the worlds collide differently with probability ρ(1 − ρ) = 2/9, and becomes three (the other directions)

the collision table

damage front, linear spreading (pulled front)

axis 0.855 per step (0.840 with Bramson's correction over t = 64–128); diagonal 0.707, the light cone itself

min over λ of ln μ(λ)/λ

causal past of a collision, traced back

each event has two incoming particles, each born at its last collision a geometric time earlier: a tree growing as e^st with s = ln(1 + ρ(1 − ρ)²) = 0.138 per step until it fills its cone; cone front 0.834 per step along the axes, 0.707 along the diagonals

renewal theory

where tree meets cone

τ ≈ 60 steps on the square torus, ≈ 35 on the strip of height 4 (crude min(tree, cone))

copying twin, mean field

synchronises when (1 − p)(1 + 2ρ(1 − ρ)) < 1, i.e. p > 4/13 = 0.308

linear damage growth

entropy of the picture, uniform gas

4,573 particles, 256 blocks of 64 bits: S_max = 13,181.0 bits; expected deficit 179.9 bits (hypergeometric), about half a nat per block

exact expectation

entropy of the picture, the disk

maximum-entropy state under all 512 invariants (a generalised Gibbs ensemble, solved to residual 10⁻⁹): mean deficit 369.7 bits over one period, swinging from 198 bits (t ≡ 16 mod 32) to 567 bits (t ≡ 0 mod 32)

maximum entropy

Check

Θ Fⁿ Θ Fⁿ x = x bit for bit, n up to 10,000, on tori, with walls and with mirrors

particle–hole symmetry: F(complement x) = complement F(x); the empty world and the full world both stand still

particle number, row and column momenta per class, and the travelling diagonal counts are exactly conserved on tori

a flipped bit's damage never vanishes and never leaves its class

collision events per site per step in equilibrium within 1% of 0.0988

the collision of a 3-site world equals P ⊗ P ⊗ P acting on its index (P the 16 × 16 site matrix)

the law of a tiny world is a permutation matrix M with Θ M Θ = M⁻¹ = M^T

any single slice of a history regenerates the whole history, forward and back

the half adder's truth table holds, and its outputs determine its inputs; a mirror loop holds a written word for 100,000 steps in vacuum

C10

averaged over bits, the number of causes of a bit at depth τ equals the number of effects of a bit after τ steps (double counting)

Word

Prediction

Confidence

Fails if

space

A pressure pulse (disk of radius 8 at 0.5 per bit in a gas at 1/3, 256 × 256, 1,000 runs) makes a ring whose peak moves at 1/√2 along the axes and along the diagonals: speed from the peak radii at t = 48 and t = 112 in [0.64, 0.78] in both, ratio axis/diagonal in [0.90, 1.10]. The ring is sharper along the diagonals (where the diagonal counts make its motion free) than along the axes (where it is damped): smaller full width at half maximum at t = 112.

0.75

either speed outside the range, the ratio outside it, or the axial ring as sharp or sharper

effect

One flipped bit in the gas at 1/3 (256 × 256, 100 runs). The level set at a quarter of the interior plateau advances along the axes at 0.84 ± 15% ([0.71, 0.97]) between t = 64 and 128, and along the diagonals at [0.62, 0.72]. The interior plateau is ρ(1 − ρ) = 0.222 per bit over both classes ([0.19, 0.26]): inside the damage the two worlds are unrelated. The damage count grows as t^α with α in [1.8, 2.3] over t = 64–128.

0.6

any of the four outside its range

spacetime

Counting the collision events in an event's causal past (512 × 512 square and 2,048 × 4 strip, 64 target events at t = 400): ln V grows linearly with slope in [0.12, 0.18] per step over τ = 10–30 in both (the exact tree gives 0.152 there); the local dimension d(τ) = Δ ln V / Δ ln τ rises above 4 somewhere in τ = 25–90 on the square and then falls; at τ = 160–200 it lies in [2.8, 3.8] on the square and [1.7, 2.5] on the strip, with the square at least 0.6 higher. The strip's peak comes earlier than the square's.

0.55

any of these

irreversibility

A Loschmidt echo in equilibrium with one bit flipped at the turnaround returns with an error equal to the forward damage of one flip after the same number of steps: mean echo error / mean forward damage in [0.90, 1.10] for n = 16, 32 and 64 (256 × 256, 50 runs each).

0.8

any ratio outside the range

entropy (and time)

The picture's entropy deficit S_max − S_B, averaged over t = 4,000–8,000 on the 64 × 64 torus: the uniform control lies in [160, 200] bits; the disk lies in [340, 430] (the GGE's 370, allowing for incomplete relaxation, which can only raise it); the disk exceeds the control by more than 100 bits. Folded modulo 64, the disk's deficit follows the GGE's phase profile with correlation above 0.9: peaks at t ≡ 0 mod 32 (ghosts of the disk), troughs at t ≡ 16, peak-to-trough within 25% of the GGE's 369 bits.

0.6

any of these; a further invariant I have missed would show as a higher level or a different profile

reality bites back

A copying twin (runs the law; each step overwrites a random fraction p of its bits with reality's; starts independent) on 64 × 64 at 1/3, 10 runs per p, 2,000 steps: there is a threshold p_c, the smallest p on the grid 0.05–0.40 at which at least half the runs reach zero error, and it lies in [0.10, 0.30]. The mean-field 0.308 is an upper bound: clustered damage wastes its own branching. At the grid point above p_c every run synchronises; two grid points below, none does.

0.6

p_c outside the range, or no sharp threshold

reality bites back

A twin that knows the law and asks yes/no questions about single bits of reality finds one unknown flipped bit among H = 16,384 (64 × 64 at 1/3), choosing each question to split its remaining hypotheses as evenly as possible. With D̄(t) the mean damage count after waiting t steps, the number of questions Q(t) lies within a factor 1.5 of H / (2 D̄) + log₂ D̄ for t = 8, 16, 24, 32 and 48 (20 runs each). At t = 48, Q ≤ 21, at most 1.5 log₂ H. After the answer, the twin equals reality bit for bit and stays equal.

0.6

any t outside the factor, or Q(48) > 21

Word

Form in the world

Kind

one

a single particle in an empty world: it moves and returns; nothing happens to it

definition

multiple

many particles: now collisions happen; the number of arrangements C(4N, k)

definition, C5

zero

the empty world stands still, and so does the full one; a hole moves like a particle

set

the world is the set of occupied slots; in a collision, which particle went where is not defined

definition, C3

point

a site is a point; the whole world is one point of a 16,384-dimensional cube; distance counts differing bits

definition

pixel

the camera: counts per 4 × 4 block; the number of worlds behind one picture

definition

vector

momentum is a sum of unit vectors; this gas conserves 256 of them

voxel

a history stacked as a block of voxels (x, y, t); one slice fixes it all

tensor

the history as a four-index array; the collision of n sites as P^⊗n

matrix

a tiny world's whole law as a permutation matrix; conservation laws make it block diagonal

space

one particle's space is a diamond; the crowd's sound is a circle inside it

time

time is iteration; a finite reversible world returns to every state; the disk's ghost returns every 32 steps

C1, P5

event

a collision, the only place the law does something

cause

a bit is a cause if flipping it changes the event; most of the light cone is not

C10

effect

one bit's damage spreads at 0.84 along the axes and at the light cone along the diagonals

P2, C4

spacetime

from the causal order of events alone: a tree at small scales, three dimensions (square) or two (strip) at large

reversibility

draw, run into smoke, turn, run: the drawing returns

irreversibility

one bit wrong at the turnaround: the drawing returns with a hole the size of the damage

entropy

the picture's entropy rises and settles where the invariants allow, not at its maximum

algorithm

one collision is a half adder; mirrors are its program

input

the particles sent into the gate

output

the particles that leave; in a reversible machine they determine the input

stream

bits in a row, moving; your own text

definition

feedback

a loop of mirrors feeds its output back to its input: a delay-line memory

bit

one particle present or absent, held in the loop; one flip

reality bites back

one bit the model does not know: copying needs a stream, the law needs 14 questions

P6, P7

Predictions, written before any simulation?

The subject?

The world?

What the gas conserves?

Theory?

Structural checks?

Predictions?

The 26 plates?

What this cannot establish?

Predictions, written before any simulation

The subject

The world

What the gas conserves

Theory

Structural checks

Predictions

The 26 plates

What this cannot establish