# Flux invariants: a rotation law the official set obeys exactly > 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. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/flux-invariants/ - Updated: 2026-07-22 - Topics: structure - Source: Flux invariants: reproduction topic with the article, the committed checker and the results JSON (GitHub) — https://github.com/raphael-anjou/eternity2/tree/main/research/topics/flux-invariants - Source: Conway and Lagarias, Tiling with polyominoes and combinatorial group theory: the boundary-word invariants this law adapts to edge matching (JCTA 1990) — https://doi.org/10.1016/0097-3165(90)90057-4 - Source: Brendan Owen digitises the piece set: the color histogram this law counts against (groups.io msg 1054, July 2007) — https://groups.io/g/eternity2/message/1054 - Source: Brendan Owen, Design the hardest puzzle: the deliberate frame/interior palette split (groups.io msg 1947, August 2007) — https://groups.io/g/eternity2/message/1947 --- Scores on this page use the edge-matching convention: a board's score is the number of the 480 internal joints whose two half-edges show the same color. A full solution is 480. The invariants below are not about scoring a board; they are about which piece rotations a valid 480-board is allowed to use, and they hold for every valid solution regardless of score. ## The one-line idea Give each of the 22 edge colors a numeric weight, with the gray border color weighted zero. Read a placed piece as a two-dimensional vector: its east weight minus its west weight on the horizontal axis, its south weight minus its north weight on the vertical axis. Now sum that vector over any block of placed pieces. Every internal seam of the block is shared by two pieces, and it enters the sum once with a plus sign from one piece and once with a minus sign from its neighbour, the same color on both faces, so the two cancel. Nothing survives the sum except the block's outer boundary. Over the whole board that outer boundary is the grey rim, weighted zero, so the total is exactly the zero vector. That is a conservation law, the discrete cousin of a divergence theorem: the flux out of every region equals the flux across its boundary, and for the whole board the boundary carries no flux at all. ## Why a quarter turn brings in the complex numbers The vector is not rotation-blind. Turn a piece ninety degrees clockwise and its north color moves to east, east to south, south to west, west to north. Work through what that does to the vector and the horizontal and vertical axes swap with a sign: the new vector is the old one turned by a right angle. In the plane, a right-angle turn is multiplication by the imaginary unit. So if you write the vector as a complex number, one quarter turn of the piece multiplies it by $i$, and the four rotations a square piece can take correspond to the four powers $1, i, -1, -i$. That single fact is what lifts a bookkeeping identity into algebra. The conservation law, written per color, says that a certain sum of Gaussian integers, one term per piece, each multiplied by a power of $i$ set by that piece's chosen rotation, must come out to zero. It is a constraint not on where pieces go but on which rotations the whole set may adopt. ## The complete family, and what is new The rotation group of a square has four characters, and the flux law is only one of them. Decomposing the piece along all four gives the complete lattice of linear, piece-intrinsic edge-matching invariants: nothing linear escapes it. The four members are a plain color census (rotation-blind, the piece's total count of each color), the Gaussian flux law just described, its complex conjugate (the same information), and a fourth, integer-valued member that couples each piece's choice of which opposite-edge pair lies horizontal to the checkerboard color of its cell. This last one is the edge-matching echo of the two-coloring obstruction that Conway and Lagarias used for polyomino tiling, the boundary-word method this whole family is adapted from. The census is old news: it is just color counting. The value of the flux law is that it is not. On the official set its rotation-blind shadow, the version you get by forgetting the $i$ and merging plus and minus, is identically zero for all 22 colors, because every color count on the board is even. In other words, color counting already knows everything the shadow could tell it, and it knows nothing more. Every constraint the flux law imposes beyond that shadow is genuinely new information a census cannot see. ## The linear algebra on the real instance Line the 22 colors up as rows and the 256 pieces as columns, and the per-piece flux coefficients form a matrix. Its rank measures how much the law actually constrains the puzzle. Recomputed on the official 256-piece set, the numbers come out as follows.
22
complex rank, full (22 of 22 colors)
40
independent real constraints
21
independent parity constraints (mod 2)
Full complex rank 22 means the law binds all 22 colors at once, with no color falling out as a free variable. Splitting the complex equations into their real and imaginary parts gives 40 independent real constraints on the rotation assignment. And reducing the whole system modulo 2, where a piece's rotation collapses to a single parity bit (a quarter turn and a three-quarter turn become the same mod 2), leaves 21 independent parity constraints, the augmented system staying rank 21 as well, so it is consistent rather than contradictory. That mod-2 system alone removes a factor of about two million from the space of rotation parities. The instance facts the law counts against reproduce exactly too, digit for digit against Brendan Owen's digitised piece set: 196 interior pieces, 56 edge pieces, 4 corners; the grey border color on 64 half-edges; five frame colors that touch only border pieces, 24 half-edges each; the remaining interior colors splitting five at 48 and twelve at 50; and every color count even, which is exactly what a perfect edge matching requires. The five official clues are all interior pieces. None of this depends on trusting a color numbering: the checker derives the frame set from the data (the colors that never appear on an interior piece) so a renumbering between the starter kit and the source list cannot fool it. ## From a law to an endgame check A conservation law that must hold for the whole board also constrains any partial board, because the pieces still to be placed must carry exactly the flux the placed pieces are missing. During a search that fills the grid piece by piece, the flux the remaining pieces owe is fixed the moment the placed set is fixed. If no assignment of rotations to the remaining pieces can supply that owed flux, the partial board is dead, and you can stop without searching its subtree. Reduced mod 2 this becomes a small linear system over the remaining pieces' rotation parities, decided by Gaussian elimination, and it is a sound endgame certificate. Sound means it never rejects a board that is actually completable: the law is a necessary condition, so a real partial always passes. What it can do is catch a mistake. Tested on a pool of framed solved 8-by-8 boards with planted rotations, injecting a single illegal rotation into the placed prefix, the check never once rejected a valid partial across 3600 trials, and its probability of catching the injected error rose with the fill fraction. | Fill fraction | Valid partials rejected | Single-rotation error caught | |---|---|---| | 0.50 | 0 of 900 | 0.6% | | 0.75 | 0 of 900 | 14.1% | | 0.90 | 0 of 900 | 88.3% | | 0.95 | 0 of 900 | 94.6% | The reproduced content here is the mechanism and the shape of that curve, not the exact percentages. The source reports a steeper curve (13%, 50%, 100% at fills 0.50, 0.75, 0.90) on a different planted family; the run above uses an 8-by-8 board at 13 colors with a uniformly chosen injected piece and a pool of 30 boards, 30 orderings each (900 per fill), so the absolute detection rates sit lower. What holds exactly, and is the point, is that the certificate is sound and that its detection probability climbs monotonically toward full as the board fills. That is precisely the useful behavior for a search: the check sharpens exactly where the tree's branching tail is most expensive, near the end, and every catch is orthogonal to color and count pruning, so it adds on top of them rather than duplicating them. ## What it is and is not The flux law is an obstruction. It can certify a partial board dead; it can never certify one completable. That is the correct and intended role for an invariant inside a branch-and-prune search, and it is shared by every result in this family. Two more honesty flags. The law constrains which rotations the piece set may use, not which cell each piece sits in; only the checkerboard member couples to position, and only through the parity of the cell, so neither member pins a piece to a place. And the natural nonlinear extension, a nonabelian boundary-word product in the style of the original Conway and Lagarias construction, does not survive in two dimensions: an interior cell has four shared edges but only two neighbours adjacent to it in any linear reading order, so at least two of its edges can never be made to cancel, and the construction collapses back to the linear flux law. Any invariant strictly stronger than these must be nonlinear and lies outside the tiling-group family. ## Where it sits This is the detailed account of one law in the [theorem sweep](/research/why/theorem-sweep), the arc that asked what could be proved about the instance rather than what score could be reached. It sits next to [ring purity](/research/why/ring-purity), the other exact law the border obeys, and it is the algebraic complement to the [permutation-code lens](/research/why/permutation-code-wall), which reads the whole board as a codeword: the flux law is a set of parity checks the rotations must satisfy, the same currency that lens is written in. The [complex theory page](/research/why/complex-theory) counts how wide the search is; this page adds a cheap, sound way to prune its endgame. Every number above is recomputed by the committed checker in the reproduction topic linked under sources: a single deterministic Rust program that loads the official instance, derives the frame colors from the data, computes the three ranks, verifies the census-orthogonality shadow, and runs the seeded endgame sweep, emitting one JSON file (committed as `results/flux_invariants.json`) that holds every figure quoted here. The instance facts and the ranks reproduce byte for byte; the certificate reports its own catch curve, sound at every fill. ## 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. - [Ring purity: the border is a closed sub-puzzle with zero slack](https://eternity2.dev/research/why/ring-purity) — 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. - [The board as a codeword](https://eternity2.dev/research/why/permutation-code-wall) — 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. - [Complex theory: counting the search before you run it](https://eternity2.dev/research/why/complex-theory) — Brendan Owen's complex theory estimates how wide the search tree is at every depth, and even how many solutions exist at all. Many in the community consider it the single most important thing to understand about Eternity II.