# The fluid frame > A perfect 60-piece border is not one rigid object. Every fully matched frame admits exactly 45 free rim exchanges at zero border cost; a third of perfect frames cannot even start the interior, and a single free exchange revives every one of them. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/lab/experiments/raphael-anjou/pipelines/fluid-frame/ - Updated: 2026-07-22 - Topics: structure, local-search - Reproduce: `just research-frame-manifold` - Source: Reproduction topic (this project): committed plan, compute crate and results behind every number on this page — https://github.com/raphael-anjou/eternity2/blob/main/research/topics/frame-manifold/article.md --- Border-first pipelines such as [CLOISTER](/research/lab/experiments/raphael-anjou/pipelines/cloister) build a fully matched 60-piece border, freeze it, and hand its 56 inward-facing colours to the interior search as hard constraints. I wanted to know whether that frozen object is really one object, starting from a simple reflex: when the interior stalls against the rim, why backtrack the border when you could exchange one rim piece? Measured on 500 fresh perfect frames, the answer is sharper than the question. A perfect border is a connected manifold with exactly 45 free exits; a third of perfect borders cannot even start the interior; and one free exchange fixes every sampled dead one at zero border cost. ## What was measured Definitions first. A frame places the 4 corner pieces and 56 edge pieces on the 60 border cells of the official 16×16 board, grey exactly outward. BB counts the matched border-to-border adjacencies along the ring, maximum 60; BB = 60 is a perfect frame. A legal exchange swaps two placed pieces of the same class (corner with corner, edge piece with edge piece); rotations are forced by the grey-outward rule, so an exchange is determined by the slot pair. An exchange is free when it leaves BB at 60. The compute crate generates its own BB = 60 frames of the official 256-piece set (a randomised depth-first placement over the 60 border cells), then measures each claim directly: 1. every legal same-class exchange and its BB cost, pooled over all frames; 2. chained free exchanges, with the free set recomputed on the current ring at every step, tracking BB, the free-move count, and how much of the ring and of the inward colour vector has drifted; 3. fillability of every interior cell touching the frame: a frame is dead when some such cell can be filled by zero of the 196 interior pieces; 4. greedy repair of each dead frame using free exchanges only. Main run: 500 independently generated frames, walk length 100, revival attempted on the first 60 dead frames. The run is deterministic; re-running the committed command reproduces the result file byte for byte. The first generated frame is viewable in the [viewer](/viewer?puzzle=official_eternity2&puzzle_size=16&board_edges=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). ## The result **Exactly 45 free moves, on every frame.** All 500 perfect frames admit exactly 45 free exchanges, never 44, never 46. No exchange ever raises BB (60 is the maximum), and the free moves compose: after 100 chained free swaps BB is still 60 and 45 free moves are still available. The count never depletes. **None of them is cosmetic.** Zero of the 45 free swaps preserve the inward colour of the pieces they touch, so every free move changes the constraint vector the border presents to the interior. A 100-step free walk visited 99 distinct rim-target vectors out of a possible 101 without ever leaving BB = 60. **Perfect is not usable.** 180 of 500 perfect frames (36 percent) had at least one interior cell that zero of the 196 interior pieces could fill, and 46 (9.2 percent) were dead at the very first cell a scan-order solver visits: the interior search dies at depth 0 despite a perfect border score. Greedy repair by free exchanges alone revived 60 of 60 sampled dead frames to zero dead cells, with BB never leaving 60. In the source study's worked example, one free swap took a framed DFS from depth 0 to depth 153 of 196. > **[Figure]** Expected vs measured (main run, 500 frames) — interactive figure. Rendered on the canonical page (link above); not shown in this markdown export. An independent-seed robustness pass (100 fresh frames, seed base 7001) agrees: 45 free exchanges on every frame, no gains, 45 of 100 frames dead (histogram 38 / 7), all 45 revived, and the walk again holds BB = 60 with 45 free moves throughout. So the punchline holds across generators and seeds: a perfect border and a usable border are different properties, and the frame can stay mutable during the search instead of being backtracked. ## What does not follow - No claim is made that walking the manifold or reviving dead frames raises completed-board scores. The source study's own A/B produced no completed framed boards, so any score effect is an open question; a producer that keeps the frame fluid during the interior search is future work. This page publishes the mechanism only. - The 45 is claimed as a property of the official piece set and replicated here on frames from a different generator. The dead-frame rate is a property of the generator's sampling: 36.0 percent on the main run, 45 percent on the independent-seed pass, so the defensible statement is a rate in the mid-30s to mid-40s percent range, not a universal 36. - The raw same-class pair count is 1546 per frame here against roughly 1458 in the source, which evidently excluded some no-op pair class. The shares and the two load-bearing rows (45 free, 0 gains) match regardless. - Frames here are unhinted, as in the source measurement; hint compatibility of frames is a separate concern and no part of this claim. ## Related - [CLOISTER](https://eternity2.dev/research/lab/experiments/raphael-anjou/pipelines/cloister) — Fix a perfect border, then search the interior with the border's edges treated as hard constraints from the very first cell. - [STAGED](https://eternity2.dev/research/lab/experiments/raphael-anjou/pipelines/staged) — Build the whole board from scratch with no pre-set frame, in stages, letting the border emerge last from whatever pieces are left. - [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 rare colors live on the frame](https://eternity2.dev/research/why/rare-color-geography) — Five of Eternity II's 22 colors appear only along the border ring, each on exactly 24 edges, never once in the interior. A structural split that shapes how every solver treats the frame.