# The puzzle has no height function

> Borrow the physicist's trick that makes crystal defects solvable and try to turn a mismatched joint into a dislocation with a conserved charge. It fails three ways: a scalar height is blind to breaks, the break set is open strings rather than closed loops, and the oriented per-color current is not conserved. Only an unsigned parity bit survives.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/no-height-function/
- Updated: 2026-07-23
- Topics: structure, search-space
- Source: No-height-function checker: article, Rust source and the per-board results table (GitHub) — https://github.com/raphael-anjou/eternity2/tree/main/research/topics/no-height-function

---
There is a borrowed idea that, if it held, would change how you attack this
puzzle. In the statistical physics of crystals a tiling carries a **height
function**: a number living on the faces that steps up or down by a fixed
amount across every edge. When the tiling is flawless the height is
single-valued. A defect is a point where walking a small loop and adding up the
steps does not return you to where you started; the leftover is a conserved
charge (physicists call it a **Burgers vector**), and defects can only ever be
created or destroyed in charge-cancelling pairs. That last fact is the prize. It
is what lets physicists use a **worm move**: a rewrite that is spread out across
the lattice but simple in height space, sliding two defects together until they
cancel.

Score a full Eternity II board by counting its matched internal joints out of
480 (the matched-edges convention used throughout this wiki; the
[records page](/research/records) puts the numbers below in context). A
mismatched joint is a **break**. If breaks were dislocations with a conserved
oriented charge, there would be a principled non-local move that lowers the
break count by walking a pair of them together and annihilating them, exactly
the kind of move that ordinary single-swap search cannot see. This page records
why that hope does not survive contact with a real board. The failure is not a
matter of taste. It is two concrete facts you can measure on any board, plus one
that needs no board at all, and the checker in the reproduction topic measures
the first two on this project's committed record boards.

## The scalar height is blind before you start

Take the most naive version first: put a single number on each cell and read the
"step" across a joint as the difference of the two cells' numbers. Walk the four
cells around any interior grid corner and add up the four steps. Each cell's
number enters the loop once with a plus and once with a minus, so the sum is
**zero**, always, for every assignment of numbers and no matter which joints are
broken. A break lives on one joint; it never changes any corner's loop sum. So a
per-cell scalar height cannot detect a break at all. It is not that it detects
them badly; it is structurally deaf to them. This one is a one-line identity, so
it is stated here rather than measured.

## The break set is open strings, not closed loops

The physicist's fix is to stop using cell values and instead put the step on a
joint **exactly when that joint is broken**. Now the loop sum around a grid
corner counts the broken joints touching that corner, taken modulo two. The
height is single-valued only if every corner touches an even number of breaks,
which is to say only if the breaks form closed loops on the grid of corners.

They do not. On every board measured, dozens of corners touch an **odd** number
of breaks. In the crystal picture those odd corners are dislocation **cores**,
the loose ends of open break-strings. A height cannot close around a loose end,
so it cannot exist globally. The count of these cores is the first number the
checker reports, and it stays large even on the best boards:

| Board (this project) | Matched edges | Breaks | Odd-degree cores | Colors with unbalanced current |
| --- | ---: | ---: | ---: | ---: |
| best matched-edges, 463 | 463 | 17 | 22 | 10 |
| a 460 board | 460 | 20 | 22 | 9 |
| a 458 board | 458 | 22 | 34 | 15 |
| an earlier 460 board | 460 | 20 | 26 | 12 |

All four are matched-edges scores, re-verified by the checker before the
topology is measured; a board whose re-scored value disagreed with its recorded
one is refused outright, so the numbers above always describe a board of known,
verified quality. The best of them sits only 17 breaks away from a hypothetical
solution and still carries 22 open-string cores. The obstruction does not soften
as a board approaches a solution, which is the whole point: there is no board
near the top where the loops quietly close.

## The oriented charge is not conserved

One more escape is worth ruling out. Maybe resolving breaks color by color
rescues a conserved charge. For a single color, orient each half-broken joint
(one where exactly one side shows that color, which is necessarily a break) as
an arrow pointing from the color-carrying cell toward its non-matching
neighbor, and add up the arrows into one vector per color over the whole board.
A genuine conserved current would sum to **zero**. It does not. On every board,
several colors have a nonzero total, listed in the last column above: between 9
and 15 of the 22 colors carry an unbalanced current, depending on the board.

The reason is structural, and it explains why no amount of cleverness fixes it.
On an actual solution every colored edge is matched to an edge of the same
color, so there are no half-broken joints and every color's current is trivially
zero. A half-broken joint is precisely an **unpaired** half-edge, one whose
partner is missing, so its arrow has nothing to cancel against. The unbalanced
sum is not a bookkeeping slip; it is the signature of the missing partner.

For orientation only, an earlier standalone notebook run measured the same core
counts on three different boards (a 451, a 463 and a 458 board) and got **28, 22
and 32** odd cores; the four boards here land at **22, 22, 34 and 26**, squarely
in the same regime, with the 463 board matching at 22 on both. The specific
colors and signs that come out unbalanced are labels tied to how one particular
board was numbered and carry no meaning of their own; only the count of
unbalanced colors, and the fact that it is never zero, is the reproduced
content. These are this project's own boards, well below the community bests of
470 under the center-clue convention and 464 with all five clues placed (again,
the [records page](/research/records) keeps that context); they are used here
because the obstruction has to be checked on a real, high-scoring board of the
official instance, not on a small generated one where it would be either vacuous
or off-instance.

## What survives is a single bit

Put the three failures together. A scalar height is deaf to breaks; the break
set is open strings with dozens of loose ends; the oriented per-color current
does not balance. Every route to a Burgers-vector height is closed. What is left
is far weaker than a height: the only conserved invariant of the break set is an
**unsigned per-color parity**, one bit per color (an element of a 22-dimensional
space over the two-element field). A bit records whether a color's count of
half-broken joints is even or odd; it has no direction and no magnitude. A bit
is not the holonomy of any height function, and no worm move can be built on it.

That is the negative result, and it is worth stating plainly because the idea it
kills is genuinely attractive: the dimer-and-defect picture is closed from the
break side, and with it the dream of a clean non-local move that walks defects
to annihilation. The physical intuition it leaves standing is more hopeful and
lives elsewhere: the right way out of a locally frozen board (see the
[rigidity wall](/research/why/rigidity-wall)) is a **correlated cluster move**,
several tiles turned together, rather than any single-site edit. That companion
story is a separate result and is not measured here.

It also fits the shape of the puzzle you can see on the boards themselves. The
breaks do not scatter; they [pile into one band](/research/why/mismatch-geometry)
and the graph of mismatched edges is always a forest, never closing a loop,
which is the same open-string fact this page proves cannot be closed. The
surviving parity bit is a cousin of the counting law behind
[why 479 is impossible](/research/why/parity-defect-floor): both are what you get
when an oriented invariant collapses to an unsigned one over the two-element
field. Direction is exactly the thing this instance refuses to give you.

## Check it yourself

The checker loads each committed board through the starter kit's board parser,
re-scores it with the canonical rule (and refuses any board whose re-scored
value disagrees with its recorded score), then makes one linear pass to count
odd-degree cores and unbalanced per-color currents. It is deterministic, uses no
randomness, runs in well under a second on a single core, and its output is
byte-stable across runs. The
[article, Rust source and results table are on GitHub](https://github.com/raphael-anjou/eternity2/tree/main/research/topics/no-height-function),
and the reproduction block on this page points at the same topic. Two caveats on
scope, which the finding states plainly about itself: this is a **qualitative**
reproduction. The exact core counts and current sums are functions of the
specific board, so they are read as "in the same regime as", never as a
bit-for-bit match to any earlier table; and the scalar-height obstruction of the
first section is a proved identity, asserted rather than recomputed.

## Related

- [Where the mismatches live](https://eternity2.dev/research/why/mismatch-geometry) — A near-perfect board doesn't scatter its few errors evenly. It packs them into one band of five rows and leaves all the rest flawless. Which band is decided by the direction the search filled the board, and you can see the mirror on the real record boards.
- [Why basin-hopping looks impossible](https://eternity2.dev/research/why/sigma-cycles) — If you can't improve a great board by polishing it, maybe you can jump to a different great board. On every record pair tested, you can't, and the structural reason why is worth seeing.
- [Why 479 is impossible](https://eternity2.dev/research/why/parity-defect-floor) — A counting argument on the official piece set forbids a score of exactly 479/480: every color's half-edges come in even numbers, and a single broken joint would leave two odd counts. The floor below perfect is 478, and at most 76 one-move near misses can surround any solution.
- [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.
- [Records & solvers](https://eternity2.dev/research/records) — Eternity II has never been solved, but nearly two decades of community effort have pushed the best board to 470/480. Who holds what, how they did it, and why some headline "480" boards are not actually the real puzzle.