# Entropy and the area law

> Eternity II has two rules: edges must match, and each piece is used once. The first is generous. All the hardness lives in the second.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/entropy-area-law/
- Updated: 2026-07-01
- Topics: structure
- Reproduce: `just research-entropy-area-law`
- Source: Fekete's subadditivity lemma, the theorem behind the entropy limit h∞ existing at all — https://en.wikipedia.org/wiki/Subadditivity#Subadditive_sequences
- Source: Shannon entropy of a source, the h(n) quantities measured here — https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

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Forget the use-each-piece-once rule for a moment and treat the 196 interior
pieces as reusable tiles. Count the valid all-matched patches and they grow
exponentially with size: there's no shortage of locally-valid ways to tile.
The matching grammar is rich, not restrictive.

We can measure that richness exactly. For a strip of width $n$, the growth
rate per cell is an entropy density $h(n)$, and the sequence decreases toward
the true two-dimensional value as the strip widens.

## The grammar's entropy, width by width

> **[Figure]** interactive: EntropyChart. Rendered on the canonical page (link above); not shown in this markdown export.

## Then the area law bites

Now put the use-each-piece-once rule back, and ask how often a random
color-valid $n \times n$ patch actually uses distinct pieces. Call it
$\rho(n)$. It collapses, and it collapses in the area, not the
perimeter:

$$
\rho(n) \;\approx\; \exp(-\alpha\, n^2), \qquad \alpha \approx 0.085.
$$

An area-law decay is brutal because area grows quadratically. The fraction of
realizable patches drops below one in a thousand at around eighty cells.

> **[Figure]** interactive: RhoChart. Rendered on the canonical page (link above); not shown in this markdown export.

## See the gap open

The same idea on the real pieces, exactly counted. Step the block size and
watch how many colour-valid blocks survive the use-each-piece-once rule.

> **[Figure]** Interactive: distinctness collapse and rho decay — interactive: EntropyScarcityLab. Rendered on the canonical page (link above); not shown in this markdown export.

## Why it matters

Eighty cells is not arbitrary. It's the size of the smallest moves that
separate the best known boards. The matching grammar stays rich up to about
that scale, then the distinctness rule collapses it. So the wall isn't in the
part that looks hard, matching colors; it's in the quiet rule that each piece
is used once, whose cost grows with area, on a board just large enough for it
to bite.

See [why basin-hopping is impossible](/research/why/sigma-cycles).

## The theorem, briefly

The per-width entropy has a well-defined limit. Joining an $n_1$-wide and an
$n_2$-wide strip side by side only adds a seam constraint, so the eigenvalues
satisfy

$$
\lambda_{n_1+n_2} \;\le\; \lambda_{n_1}\,\lambda_{n_2}.
$$

Taking logs makes $\log \lambda_n$ subadditive, and Fekete's lemma gives the
limit as an infimum, which is exactly why the curve above decreases:

$$
h_\infty \;=\; \lim_{n\to\infty}\frac{\log_{10}\lambda_n}{n} \;=\; \inf_n \frac{\log_{10}\lambda_n}{n}.
$$

The upper bound is the purely horizontal rate: ignoring vertical constraints
only adds patches, so

$$
0 \;<\; h_\infty \;\le\; \log_{10}\lambda_H \;=\; 1.6645,
$$

with $\lambda_H = 46.18$ the spectral radius of the horizontal
color-compatibility matrix. Positivity holds because the grammar supports
exponentially many chains, so the density is strictly between zero and
1.6645, measured near 0.67.

## Related

- [Tuned to the hardness peak](https://eternity2.dev/research/why/phase-transition) — Eternity II uses 22 colors. They split 17 interior to 5 frame-only, and that 17 is exactly where this kind of puzzle is hardest to solve.
- [Why a faster computer doesn't help](https://eternity2.dev/research/why/prune-vs-speed) — The single most important idea in hard combinatorial search: shrinking the space you search beats searching it faster, by an exponential margin. Eternity II is engineered so you can barely shrink it at all.
- [Forbidden patterns](https://eternity2.dev/research/why/forbidden-patterns) — Almost every small patch of pieces you could build is impossible. For a 2×2 square, 99.72% of the ways to place four pieces can never be made to match.
- [Solution counting: measuring what you cannot find](https://eternity2.dev/research/build/analysis/solution-counting) — Nobody has ever seen a full Eternity II solution, yet the community knows, to within a factor of two, how many exist. This page is the story and the craft of that number: exact censuses on small boards, the expectation formula and its twenty-year convergence on 14,702, and the culled-search estimates that were trusted only when four independent runs agreed.