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.
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 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.
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 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.
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 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.
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) 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 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: 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.
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, 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.