Theory

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.

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.

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.

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.

Five of the 22 colors never touch the 196 interior pieces. The piece list forces every valid solution to spend all 120 frame half-edges on the border ring: a self-contained sub-puzzle at exactly zero slack (120 = 120), an Eulerian circuit on five vertices, coupled to the interior through just 56 inward edges.

Weight every edge color and read each piece as a signed vector, east minus west on one axis, south minus north on the other. Summed over any region the interior seams cancel and only the boundary survives, so the whole board totals zero. A quarter turn rotates the vector by a right angle, making the law algebra in the Gaussian integers.

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.

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.

Fill a board in any fixed order and the top three quarters go in freely while the difficulty piles into whichever band you finish last. Across forty generated boards the entire leftover sits in the bottom half every single time; shuffle the fill order and it scatters, so the hard region is made by the sweep, not hidden in the board.

Sum, over any visit order of the 16x16 board, the number of already-placed neighbours each cell faces at the moment it is filled: the total is exactly 480, for every order. A fill order cannot add restriction; it only chooses when each restriction binds. What separates orders is immediacy, the distance between a decision and its refutation, and only the extremes of that ranking are properties of the puzzle rather than of the engine.

Read a full board as a codeword whose 480 internal joints are parity-like checks, and the matched-edge score becomes 480 minus the number of failed checks. It is a clean lens with one load-bearing identity underneath it, and it is worth being exact about what the coding view buys and what it only renames.

The border ring is the most constrained part of the puzzle, so a different strong border ought to pin a different high interior. It does not. Many distinct fully-matched borders, each completed by one fixed interior producer, give tops that are near-maximally different from each other yet uniformly low, none near the record band. The border diversifies the board without predicting its ceiling.

A solved board hides a simple bookkeeping law: every colour the border hands to the interior, the interior hands straight back. Break it and you know instantly the board is wrong; obeying it, though, guarantees nothing.

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.