# The 470 wall: a phase boundary, not an engineering limit

> The community plateau in the high 460s reads as an entropic phase boundary of the instance, not a limit of solver engineering: exact computation on the official set gives a constraint density near 0.0094, an annealed landscape that collapses above 470 and crosses 1 only at 480, and an expected 10 to 20 mutually near-orthogonal perfect solutions. The instance-side numbers are exact; the 16x16 overlap-gap picture is a stated conjecture.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/the-470-wall/
- Updated: 2026-07-22
- Topics: structure
- Reproduce: `cd research/topics/the-470-wall/compute && cargo run --release --bin the_470_wall > ../results/landscape.json`
- Source: Achlioptas & Coja-Oghlan, Algorithmic barriers from phase transitions (FOCS 2008): solution clustering and where local algorithms stall in random CSPs — https://arxiv.org/abs/0803.2122
- Source: Gamarnik, The overlap gap property: a topological barrier to optimizing over random structures (PNAS 2021) — https://www.pnas.org/doi/10.1073/pnas.2108492118

---
The best public boards have clustered in the high 460s for years, and the
ceiling stands at 470 of 480 (the record history lives on the
[records page](/research/records)). The question this page asks is whether
those last ten points are an engineering problem, something a better
backtracker or a sharper heuristic will eventually claw back, or a property
of the instance itself. Treating the official piece set as one member of a
planted random ensemble gives a quantitative answer, and the answer points at
the instance: the supply of boards a search can actually reach collapses
right where the community stopped.

The argument has two layers with very different standing, and I keep them
separate throughout. The numbers computed on the real piece set are exact,
and the compute step behind this page reproduces them bit for bit. The
structural picture that interprets those numbers at 16×16 is a conjecture,
supported by exhaustive enumeration of small instances tuned to the same
parameter, and it is labeled as such below.

## The bag, measured exactly

The 256 pieces split into 4 corners, 56 edge pieces, and 196 interior pieces.
Setting aside the 64 gray rim half-edges leaves 960 colored half-edges, which
pair into the 480 internal adjacencies of a filled board. The accounting is
tight: $960 = 2 \times 480$, with no slack anywhere. Scores on this page
count matched internal adjacencies out of 480, the same matched-edges
convention as the community ceiling; nothing here is a claim about the strict
five-clue track.

The color economy then splits into two subsystems that never talk to each
other. The frame ring, the cycle of 60 joints between border pieces, uses
only colors 1 to 5, each appearing on exactly 24 half-edges; the probability
that two uniformly drawn frame half-edges agree is
$p_f = 5 \cdot (24/120)^2 = 0.200$. The interior subsystem covers the other
420 joints on colors 6 to 22 (five colors at 48 half-edges, twelve at 50),
giving $p_i \approx 0.0588$, almost exactly $1/17$. These two probabilities
carry the whole analysis.

One more exact check is worth recording: the real bag contains zero tiles
that repeat under rotation. Against a matched random null this appears to be
the one statistically significant fingerprint of the design pass; the
null side of that comparison still needs its own generator, so only the
real-set side (exactly zero) is verified here.

## One number places the regime

The parameter that positions Eternity II inside its ensemble is the
constraint density: the expected number of tiles that fit a fully constrained
interior cell, one whose four neighbours are already placed. With 196
interior tiles, 4 rotations each, and a per-edge collision probability of
0.0589 for a random interior tile edge,

$$\mu = 196 \cdot 4 \cdot 0.0589^4 \approx 0.0094.$$

A fully surrounded hole admits about one candidate in a hundred. That is far
below one, which places the instance deep in the rigid regime of planted
constraint-satisfaction ensembles: the regime where theory says the solution
set shrinks to isolated, well-separated points, and where local algorithms
provably stall short of them
([Achlioptas & Coja-Oghlan 2008](https://arxiv.org/abs/0803.2122),
[Gamarnik 2021](https://www.pnas.org/doi/10.1073/pnas.2108492118)). The
number itself is an exact function of the real color counts; what the regime
implies at this size is the conjectural layer, taken up below.

## The annealed score landscape

The instance-side centerpiece is a first-moment count: how many board
configurations that are *uncorrelated* with the planted solution reach a
given score? The base count of class-respecting placements (corners in
corners, edges on the rim, interiors inside, free rotations for interior
tiles) is $W_{\text{geom}} = 4! \cdot 56! \cdot 196! \cdot 4^{196} \approx
10^{559.9}$. The score of a random such configuration is the sum of 60
Bernoulli($0.200$) frame indicators and 420 Bernoulli($0.0588$) interior
indicators, and an exact log-space convolution of those 480 variables gives
the full landscape. A uniformly random configuration scores $36.7 \pm 5.7$.

| score | uncorrelated configurations at this score (log10) |
| ----: | ------------------------------------------------: |
|    37 |                                              558.8 |
|   200 |                                              462.8 |
|   400 |                                              186.6 |
|   460 |                                               59.6 |
|   470 |                                               33.1 |
|   480 |                                               +1.28 |

Two things stand out. First, the supply of high-scoring uncorrelated boards
stays astronomical remarkably far up: about $10^{60}$ configurations at
score 460 and still about $10^{33}$ at 470. Second, the count crosses 1
essentially at 480 itself: the expected number of perfect placements
uncorrelated with the planted board is $10^{1.28} \approx 19$. The working
model this prices out is a solution set of order 10 to 20 perfect boards,
mutually near-orthogonal and orthogonal to the planted one, at the top of an
entropy curve that only just clears zero.

## Exact below, conjecture above

Everything above this line is an exact computation on the official set: the
class counts, the 960-half-edge economy, both collision probabilities, $\mu$,
$W_{\text{geom}}$, and every row of the landscape table. The repro command on
this page regenerates all of it from the shared engine in under a second,
into `results/landscape.json` in the topic folder.

What the landscape does not tell you is how those rare high scorers are
*arranged*: whether the entropic mass connects to the perfect boards or
whether an empty gap separates them. That question I can only answer exactly
on small instances. In the source arc, exhaustively enumerating planted $n \times n$ boards for
$n$ up to 7, with the color count tuned so the constraint density matches,
shows a clean trend: at loose $\mu$ the overlap-with-planted histogram of the
solution set is continuous, and as $\mu$ drops toward E2's 0.009 it turns
bimodal and then collapses. At the matched point ($n = 5$, 11 colors,
$\mu = 0.009$) the enumeration finds the planted solution, a cluster right
next to it, a large family at overlap zero, and an entirely empty band in
between.

> **Where the conjecture starts**
>
> The 16×16 statements (a solution set of 10 to 20 near-orthogonal perfect boards, an empty overlap gap below them, and the 470 wall as the visible edge of that gap) are extrapolations of the small-instance trend along the constraint-density parameter. They are conjecture, not measurement: no feasible computation checks them directly at full size. The landscape table and every instance-side number on this page are exact; the gap structure at 16×16 is the part you should hold as a working model.

## Reading the wall

Put the exact layer and the conjectured layer together and the community
plateau stops looking like a tooling deficit. A heuristic that climbs the
score landscape is drawing from the entropic band, and the band is deep: with
$10^{33}$ uncorrelated configurations still available at 470, reaching the
high 460s is cheap in a precise sense, and solver engineering has been
harvesting that band for years. Past it, the supply thins by thirty-odd
orders of magnitude over ten points of score, and if the overlap-gap
extrapolation holds there is nothing in between to climb through: the missing
ten points are the width of an empty region separating the last entropic
boards from a handful of isolated perfect ones. That is the overlap gap
property in its textbook role, a topological barrier that stable, local
search cannot cross regardless of implementation quality.

This reading agrees with what we measure elsewhere on the wiki: the
[rigidity wall](/research/why/rigidity-wall) finds record boards frozen into
isolated local optima with no gradient outward, which is exactly how the
bottom edge of a gap should feel from below. It also sharpens what "progress"
would have to mean. More speed and better ordering buy entropic points, and
those run out around 470 by the table above; anything that crosses the gap
has to inject correlation with an actual perfect solution rather than climb
the score function. The statements about the instance that hold at proof
grade, as opposed to the working model on this page, are collected in the
[theorem sweep](/research/why/theorem-sweep).

## Related

- [The theorem sweep: thirteen structural laws](https://eternity2.dev/research/why/theorem-sweep) — One research arc, thirteen families of structural theorems: ring purity, the 479 parity floor, the 470 wall as a phase boundary, flux invariants, the entropy area law, and the impossibility results that price every standard shortcut. This page is the map.
- [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 count of around 17 lands near where this kind of puzzle is hardest to solve (the transition is a band, not a single integer).
- [The rigidity wall](https://eternity2.dev/research/why/rigidity-wall) — Every record board we have is frozen in place. You cannot nudge your way from a great board to a perfect one, and we can prove it.
- [Designed to be unsolvable: the recipe](https://eternity2.dev/research/why/design-recipe) — Eternity II follows a recipe for the hardest possible edge-matching puzzle: compact shape, no symmetric or duplicate pieces, split palettes, flat frequencies, one expected solution. The community reverse-engineered every ingredient in the launch year.