# CAS > Place a perfect border ring first, then solve the board inward, shell by shell, each ring as one assignment over the pieces that remain. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/lab/experiments/raphael-anjou/pipelines/cas/ - Updated: 2026-07-22 - Topics: construction, local-search - Reproduce: `just research-cas-annular` - Source: Reproduction topic (this project): committed plan, compute crate and per-frame results behind every number on this page — https://github.com/raphael-anjou/eternity2/blob/main/research/topics/cas-annular/article.md --- CAS (concentric annular solving) builds the board the way an onion grows: lay a perfect border ring, then solve inward, shell by shell, until the four centre cells close the board. I asked whether that outside-in schedule could be a record path. It came back with a clean no and a useful yes: from any perfect frame the pipeline plateaus in a narrow 429 to 437 band, and from that exact starting state it beats a destroy-and-repair continuation on every single frame. ## How it works The 16×16 board is 8 concentric rings of 60, 52, 44, 36, 28, 20, 12 and 4 cells. CAS fixes the outermost ring first as a perfect 60/60 frame: all 60 border cells filled with border pieces, every rim edge grey, all 60 adjacencies inside the ring matched. It then solves each inner ring in turn as one assignment problem over the pieces still unused, maximising matched edges against the ring outside it plus the matches inside the ring itself. Once a ring is placed it is frozen; the search never revisits it. The original runs solved each shell with a MIP, 25 to 45 seconds per frame. The committed reproduction swaps that for a deterministic width-1024 per-shell beam, which lands in the same band, and adds a comparison arm: pin the same frame, leave the interior empty, and hand that state to a [destroy-and-repair local search](/research/build/local-search/local-search-alns) instead. ## The result Twenty enumerated perfect frames, twenty scores between 429 and 437 matched edges out of 480 (mean 432.8). The source audit, on its own 20 frames, sat at 430 to 436 with mean 432.4. The band and the mean reproduce to within one point on each edge, and the frame does not matter: every 60/60 frame lands on the same plateau. The best reproduction board scores 437; it is checkable [edge by edge in the viewer](/viewer?puzzle=official_eternity2&puzzle_size=16&board_edges=abdaabjbaeqbadteaftdafgfacrfabpcafubabjfafhbafqfadofadhdaendaabedgdajiggqphitwhptgiwgkogrphkpjppumhjjvomhnlvqgtnouvghmwuniombafjdpcagnnpwlmnhuqlisouoqwshwmqppvwhjqpoqtjlvmqtkuvvulkwswuommsfaemcpcansvpiujsqrluonjrkpgnmonpvmjoqgqmtrkgmhmrunhhlktnwhqkmorheabocnfavhunjlthljmlhukjgsquntksjkqtqmokkigmmhgihruhtrgrqpqrrhkpbachfoeaurhotusrgolukuwopmmukrvmqrtropprgmspgsjwuvusgtqvqvntkokvcacoeteahustsgrulorgwuiogvvlvitvtvphpsuvskpphvrkuwovqlhwnkolkgikcadgencaspwnrilpriliiklqvwvkuiwqooiqijjrprsjrmorinjmhsknoiisiwgidafwcqeawinqliwillkilwolvwlwwmulijjmqwlrtqqwosnqjtrukmttiommgllofaelepbanqrpwoiqklwooupllsnhuhwklsuhhwusqngwnnsnrqunttlqmrgtlslreaesbgbaringigqiwwvgpmiwnnvjvijnuksivrtknkvrssskujoslijjgpnilulpeaeubteanpgtqgmpvgsgisjgvntorkmnsmwktrsmvmtrtplmoslgjuhsnkoulqjkeafqeibagtgimsutsphsjnmptmqnmrwmwpkrstoptpjtlsnptgwshhlgouqnjuoufafubmdagmtlujvmqovjmnloqoqnwmsokqhmoriqjkgrntkkwvwtlrwvqvproknvfackdsdatjisvvijvlwvlkplqjskshvjhrphijurgwvjkrvwwhtrwjhhpjijnthjcadtdcaaidacidadweadpcaesbacveabpdaeufadvcafvbactfabhcafrbachcabdaac&hints=135.0-210.1-34.1-221.2-45.1). > **[Figure]** Expected vs measured (20 CAS frames, 8 baseline frames) — interactive figure. Rendered on the canonical page (link above); not shown in this markdown export. The second half is the head-to-head. Started from the same pinned frame with an empty interior, the local-search continuation loses every single time. In the source measurement (full alns_only, 60 seconds per frame) it reached 385 to 398 and lost all 8 overlapping comparisons by 34 to 49 points. The reproduction's lighter proxy baseline does better in absolute terms (411 to 419) and still loses every frame, by 13 to 23 points. The gap's magnitude is not comparable across the two setups; its direction and its every-frame consistency are. ## Why it stops The plateau is not a tuning problem, and it is not an objective gap either: all 480 edges are in the CAS objective. The mechanism is piece starvation. Greedy outer shells lock in pieces by how well they fit outward, and by rings 5 to 7 the remaining pool simply no longer contains pieces whose colour profiles fit the inward-facing constraints. It is the ring-scale version of [piece theft](/research/why/piece-theft): a scarce piece spent early, rings before the cell that needed it. The source volumes tried the obvious rescues, and both confirm the diagnosis. A CAS-ALNS hybrid scored 418, worse than CAS alone; an ALNS refine pass on top of a finished CAS board reached 437 to 439, a lift of a few points that stays far below what other pipelines in this notebook reach from scratch on the same hardware. For where the community bar sits, see the [records page](/research/records). ## What does and does not follow CAS is two things at once, and both halves are the finding. As a record path it is refuted as a greedy schedule: the outside-in order starves its own endgame, from any frame. As a sub-problem tool it is the strongest option I have measured for one specific state, a perfect frame plus an empty interior. Destroy-and-repair operators expect a full board to mutate; handed an empty interior they must first build one, badly, while CAS is exactly the constructive tool for that state. Which algorithm is right depends on the state you start it from; composition matters as much as the components. [CLOISTER](/research/lab/experiments/raphael-anjou/pipelines/cloister) is the sibling experiment that keeps the pinned frame but drops the ring-by-ring schedule. Three caveats bound the claim. The reproduction pins the five official clue pieces; the source pages do not record whether the original runs did. All five clues sit in the interior, so the frame arm is unaffected, and pinning can only make the reproduction slightly conservative, the safe direction for a plateau claim. The baseline arm is a light proxy, so only the direction and the every-frame consistency of the gap carry over, not its size. And the numbers are properties of the official 16×16 piece set: the mechanism lives in rings 5 to 7, which an 8×8 board (4 rings) never reaches, so small-board tests can neither confirm nor refute the claim. ## Reproduce `just research-cas-annular` re-runs the whole thing from scratch: it enumerates the frames by seeded border DFS, runs the 20-frame CAS arm and the 8-frame baseline arm, and regenerates the committed results file with per-frame scores, deltas and a viewer URL for every board. ## 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. - [Piece theft, where solvers die](https://eternity2.dev/research/why/piece-theft) — A solver fills a few rows for free, then hits a wall in the middle of the board. Here's the mechanism: a scarce piece spent in the wrong place, rows ago. - [Local search and ALNS](https://eternity2.dev/research/build/local-search/local-search-alns) — Destroy part of a board, rebuild it better, and let the algorithm learn which demolitions pay. Adaptive large-neighborhood search is the most reliable polisher this project has, and the cleanest demonstration of the wall where polishing ends. - [Dead ends](https://eternity2.dev/research/build/dead-ends) — Approaches we tried that look promising and don't move the needle on Eternity II, written down with what we found so you can spend your time elsewhere.