# The frame is not the basin: a different border does not open a different high board > 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. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/frame-is-not-the-basin/ - Updated: 2026-07-22 - Topics: structure, search-space - Reproduce: `just research-frame-is-not-the-basin` - Source: Frame is not the basin: reproduction topic with the article, the committed producer and the two results JSON files (GitHub) — https://github.com/raphael-anjou/eternity2/tree/main/research/topics/frame-is-not-the-basin --- There is a tempting shortcut for Eternity II. The border ring is the most constrained part of the board: sixty cells over only five ring colours, filled by the four corner pieces and the fifty-six edge pieces (the [ring-purity page](/research/why/ring-purity) shows why those five colours never leave the rim). If a fully-matched border is what pins the top of the board, then a *different* fully-matched border should pin a *different* high interior, and building several distinct strong borders would open several distinct high basins. Border-first diversification would be a lever for scores. This page is the measurement that closes that door. Generating many structurally distinct fully-matched borders and completing each one with the same fixed interior producer gives tops that are near-maximally different from one another and yet all land in the same low band, nowhere near the record region. The border is a strong *diversifier* of the resulting board and a useless *selector* of its quality. Frame identity does not predict the interior ceiling.
28
distinct fully-matched borders, two seed bases
~190 / 196
interior cells that differ between any two tops
441
best board, strict 5/5, still 14 short of 455
Every score on this page is under the strict five-clue convention (all five official clues obeyed, matched non-grey adjacencies out of 480); the five clues sit at interior cells, none on the ring, so every completion here is a legal strict-5/5 board and the border is hint-unconstrained. For where these numbers sit against the community and the notebook records, see the [records page](/research/records). ## The frame is a loose object The reason to doubt the shortcut is structural. The border ring is a self-contained cyclic edge-matching problem over five colours with balanced supply, so fully-matched borders are astronomically abundant: the frame alone is long known to admit more than $10^{15}$ arrangements. Structurally distinct ones are trivial to produce, and they couple to the interior through a single low-information channel: sixty inward-facing colours, a boundary word the interior has to meet, over a 196-cell, 22-colour interior. That channel is far too thin to determine the interior. So the hope that a different frame pins a different high interior rests on the frame carrying information it does not carry. The experiment turns that structural argument into a direct test. ## What was measured On the official 256-piece set with the five official clues pinned: 1. generate N structurally distinct fully-matched borders from scratch (a randomised depth-first placement over the sixty border cells, grey outward, every ring adjacency matched, deduplicated by exact ring equality); 2. freeze each border and fill its 196 interior cells with **one** fixed producer at a fixed budget: a break-tolerant beam that fills the interior row by row, scoring each candidate by matched-minus-mismatched edges against its placed neighbours (the frozen border included) and pruning to a fixed width with a seeded tie-break; 3. report the resulting full-board score band, the pairwise interior tile-Hamming distance between the best completions, and whether any border reaches the record band. Two questions, and only two: **are the tops distinct?** and **are the tops high?** ### The main run Sixteen structurally distinct fully-matched borders, interior beam width 128, three seeds per border, best kept: | Quantity | Result | | --- | --- | | Distinct fully-matched borders generated | 16 (from 17 depth-first attempts) | | Top score band (min / median / max) | 434 / 437 / 440 | | Any border reaches the record band (>= 455) | no | | Gap from the best border-first top to 455 | 15 | | Pairwise interior tile-Hamming (of 196 cells) | 188 to 191, mean 190.4 (120 pairs) | | Best board, strict 5/5, border still fully matched | yes and yes (440, 40 breaks) | The two load-bearing rows are the third and the last-but-one. **No border reaches the record band**: the best of the sixteen scores 440, still 15 short of 455 and about 20 short of the notebook's strict-5/5 high. And the completions are **near-maximally distinct**: between any two of them, 188 to 191 of the 196 interior cells differ. The borders diversify the board almost completely and select its quality not at all; every one of the sixteen lands inside a six-point window, 434 to 440. The best board of the run, a legal strict-5/5 completion of a fully-matched border, is viewable from the committed results file in the reproduction topic. ### The robustness pass Because the whole point is that frame identity does not matter, the run has to show its result is not an accident of one seed base. A second, independent pass (twelve borders, a different seed base) agrees: band 434 to 441 (median 438), best 441 still 14 short of 455, interior tile-Hamming 187 to 191 (mean 190,3), and the best board again a strict-5/5 completion of a fully-matched border. Two independent seed bases, 28 distinct borders in all, and not one of them crosses 445, let alone the record band. ## Why a stronger interior only sharpens it An earlier notebook pass ran this same test with a weaker, hard-fit interior that stalled in the 250s and never approached the record band. This reproduction deliberately swaps in a *stronger* interior producer, the break-tolerant beam above, which always fills all 196 cells. Its absolute band is therefore higher, the mid-430s rather than the 250s. That does not soften the finding; it sharpens it. Even with a much better interior solver, no distinct border reaches the record band, and the borders remain near-maximally distinct. A stronger interior cannot make the border informative, because the border is a downstream, low-entropy shell hanging off the interior, not the thing that pins it. The band is a property of the interior solver; the negative is a property of the frame-to-interior coupling, and it is producer-independent. ## What this closes, and what it does not The result is a rigidity statement about the border. It closes border-first diversification as a route to higher scores: a different fully-matched border gives a different board, but not a higher one, so there is no gradient to climb by walking the space of borders. Where the high-scoring information actually lives is the interior of the top rows, and that is exactly the region the [rigidity wall](/research/why/rigidity-wall) describes as a frozen island. The frame does not pin the top; the top pins itself, and the border hangs off it. The seam bookkeeping that couples the two, the exact colour balance across the border-interior interface, is the subject of the [border-balance page](/research/why/border-balance). This is a single-configuration negative, scoped plainly. It is established purely from borders generated from scratch and completed by one fixed producer, which is the robustness form the original study asked for. It does not attempt the harder control of freezing a record board's own border and completing it, which needs a board that is not part of the public starter kit; the negative stands without it. And the absolute band is the interior solver's, not a claim about any exact digit: the claim that carries is the record-band gap, and that reproduces with room to spare. Every number above is recomputed by the committed producer in the reproduction topic linked under sources: a Rust program that loads the official instance, generates its own fully-matched borders, freezes each and completes the interior with the one fixed break-tolerant beam, and emits two JSON files (the main run and the independent-seed pass) holding every figure quoted here. ## Related - [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 border balance](https://eternity2.dev/research/why/border-balance) — 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. - [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.