Which wall stops which method

This is the niche-picking tool. If you want to choose where to spend effort, read down the columns: pick the wall you find most interesting, and the table hands you every method that has attacked it and the exact score where it stopped. For a broader survey by method family, see the map of every known approach; for the angles still open, the open problems board; and for the approaches that provably do not move the score, the dead ends.

Every cell of the map below is grounded in this project's published work. The four walls themselves are corroborated by the published literature; the experiment results are this project's own work, each written up on its own page; they are not claimed to be externally verified.

What each wall rests on#

Not every wall is the same kind of evidence. Some are exact, deterministic computations committed to this repository; one is externally replicated by several independent methods; several are real measurements taken on the one Eternity II board and its record set. That is a legitimate base, but a reader deserves to know which is which before leaning on any single line. The table below reads every structural claim in this section down to the strongest form its evidence actually supports, the tier of that evidence, whether any outside party has reproduced it, and the one caveat to carry.

The four tiers

Proven, external. A theorem or an exact computation an outside party has stated or replicated, and the page cites that party.
Proven, internal. An exact, deterministic computation done here, with the code and result committed under research/topics/ and reproducible from the page's own command.
Measured, single instance. A real measurement, taken on the one Eternity II board (or one E2-like board), with no replication. True, and not a defect on its own, but it is one data point.
Conjecture. The page itself frames the claim as unproven in general.

A wall is not marked down for being measured on a single instance. It is marked down only where a lower tier is dressed as a higher one, and none here is.

Wall / claim	Strongest supported form	Tier	Outside replication	The one caveat
Hardness peak	17.14 interior colours follow from the one-expected-solution criterion; 17 sits near the search-hardness peak for framed edge matching	Proven, external	Yes: Owen's 1947 derivation; Ansótegui, Béjar, Fernández & Mateu confirm the peak	"One expected solution" is a first-moment average assuming independent edge colours; it locates the peak, it does not count solutions
Complex theory / the funnel	A calibrated first-moment estimator of tree width and solution count, right to within about a factor of two everywhere it could be checked	Conjecture (page badge), on external replication	Yes: Owen's model; McGavin's C port and 10×10 solve, inside prediction	It is an average, blind to whether counted partials are genuinely distinct; a tool for ranking scan orders, never a true count or a bound
No forced moves	An exact exhaustive count over the official interior set: no interior piece is ever forced to a single right-hand partner (73–137 partners each)	Proven, internal	No external replication; the count is this project's own, committed and reproducible	It counts pairwise compatibility on the full piece set, not candidate counts inside a live partial board; a property of the pieces, not a proof search never narrows
Forbidden patterns	Exact exhaustive counts: 38.96% of pairs, 83.26% of L-trominoes, 99.72% of 2×2 squares are infeasible over distinct interior pieces	Proven, internal	The 2×2 counts are this project's committed exact result; the 20 void corner-pairs are a cited 2008 community observation	The "forbidden-patch count tracks distance to a solution" reading is a heuristic signal, not a proven monotone distance
Entropy area law	The block counts A(n)/B(n) are now exact in-repo through n=3 (B(2)=4,059,952 cross-checks the sub-grid table), giving an in-repo exponent α≈0.044; the entropy limit near 0.67 and the collapse point are extrapolations	Proven, internal (the block counts) + measured (the fit)	Partial: Fekete's lemma and Shannon entropy are the external theorems that make the limit well-defined; the α fit and collapse scale are this project's own	The theorem guarantees a positive limit exists; it does not fix its value. The exponent is fit over n≤3 and its per-block value is still rising, so the large-patch collapse point is an extrapolation past the counted widths
Rigidity	On the public record boards, no rearrangement of a board's own pieces inside a halo out to four cells closes any mismatch (SAT: UNSAT), now committed	Proven, internal for the SAT halo; the deeper MIP table is proven for the regions closed but carried from an off-site paper	Yes, the best-replicated wall: Millilaw's freeze and SAT-residual tests reproduced here with a positive control; benj39100's GPU-annealing hard core reaches it from a third method	The committed site evidence is the SAT halo to radius 4 on five boards (a few radius-4 instances time out, recorded open). The MIP proof table and the board-wide ≤476 bound are sourced to the off-site paper, and the all-boards statement is a conjecture
Sigma-cycles	Over every ordered same-piece-set pair of the bundled boards (246 pairs, 1,154 large loops), every proper prefix of every large loop scores strictly worse than its start, across all 54,238 partial applications	Proven, internal (a population statement over the bundled set)	No external replication; computed here against McGavin's 469 and the bundled record and project boards	The count is exact over the boards shipped; the "every sub-cycle is worse" property is still not proven to hold for every board that could exist. A strong population observation, not a general theorem
Mismatch geometry	On the record and project boards, residual mismatches cluster in one row-band that flips with build direction; a hole-count objective lands leftovers in the same band	Measured, single instance	Partial: the scan-direction flip is this project's reading; Verhaard's seven-hole board and Zamofing's top-band residual are cited same-direction observations, now verifiable in the extended archive	A pattern across a handful of boards with a mechanistic story, not a proof
Piece theft	An exact count of (north, west) demands over the interior set: most have one to three suppliers, and a specific number have exactly one	Proven, internal (the counts); measured (the "where solvers die" mechanism)	No external replication; Régin 1994 is cited as the all-different theory that names the mechanism	The scarce-supplier counts are exact; "this is where real solvers die" is a mechanism illustrated on the instance, not a measured failure rate across solvers
Hint geometry	On one 16×16 E2-like puzzle, scattered hints solve it in minutes where piled contiguous rows do not, because the work lives in the back half of the fill	Measured, single instance (an E2-like board, not the official puzzle)	Yes, cleanly attributed: McGavin's 18-hint solve and 41-billion-node tree; Joe's depth statistics	Not the official puzzle, whose five fixed clues differ. The depth figures are one backtracker's sample
Border balance / NS-1	An exact necessary condition; the four known full solutions satisfy it; a positive deficit certifies infeasibility. The interior-to-interior share of remaining errors is now exact: 86.9% across the nine bundled 469-class boards, none border-to-border	Proven, external (the condition), proven, internal (the error split)	Yes for the condition, cited to 2007–2022 community statements; the error split is this project's committed recount	The condition is necessary, never sufficient: deficit zero proves nothing, and the invariant is blind to the interior-to-interior majority of remaining errors. The prune yield is a single-instance measurement, not yet a committed benchmark
Rare-colour geography	An exact count over the official set: the five border colours never appear on an interior edge	Proven, internal	No external replication of the count; the 17+5 design intent is cited to Owen	Correctly framed as structural (grey rim, separate colour pools), not a scarcity trick; the "rare" label is an artefact of fewer border edges
Design recipe	The community reconstructed a coherent hardest-puzzle recipe, ingredient by ingredient, each sourced to a launch-year post	Conjecture (page badge), each ingredient externally sourced	Yes, densely: Owen's derivations and measurements, the design-space census, the provenance chain to Selby and Riordan	A reconstruction of design intent, not a statement the designers published; the page badges it conjectured
Complexity framing	Edge matching is NP-complete as a family; a single fixed board is a constant, not a problem; E2's difficulty is empirical over a ~10^557 space	Proven, external	Yes: Demaine & Demaine 2007 for NP-completeness; Ansótegui et al. for the empirical case	The category distinction is load-bearing: NP-completeness caps what general solvers can promise, it says nothing about this board
Prune versus speed	The compounding argument is exact arithmetic; on E2, legal pruning often costs more than the subtree it removes	Proven (the arithmetic), measured (the "pruning does not pay" verdict)	Partial: the principle is self-contained; the community verdict is cited to McGavin and 95A31, now verifiable in the extended archive	The demo tree numbers are illustrative, not a measurement of any real solver, and the page says so

Three shifts since this map was first drawn are worth calling out, because they move claims up a tier. Sigma-cycles was a three-pair observation; it is now an exact statement over every same-piece-set pair of the bundled boards, 54,238 partial applications with zero exceptions. The border error split is now an exact recount (86.9% interior-to-interior) rather than an off-site figure. The entropy area law's block counts are now computed in-repo and reproduce from the page's own command, with the exponent fit over the exactly-counted range. And the four community citations that once sat past the archive snapshot now validate against the extended export, so the rigidity, mismatch-geometry, and prune-versus-speed corroboration passages rest on sources a reader can check.

The reading across the table: only rigidity carries three independent methods (SAT, MIP, annealing) reaching one verdict. The remaining walls are exact single-project computations or single-instance measurements on the one board that matters. Read "corroborated by the literature" as meaning the phase transition and the missing gradient have been reproduced elsewhere, not every wall.

The four walls, in one line each#

Every one of them is a version of the same statement: there is nothing local to prune on.

No forced moves. Every interior cell keeps 73–137 legal pieces, so the branching factor never collapses.
The hardness peak. With ≈17 interior colours the puzzle sits at the phase transition: about one expected solution, the worst place to search.
The area law. Genuinely-distinct partial boards collapse as the filled area grows, but no local score can see that global fact.
Rigidity. Records are locally frozen; the step to a better board is one giant, indivisible swap with no gradient to follow.

The map#

Columns are the walls; rows are the methods, best score first. A filled mark (●) means the method is fundamentally working against that wall. The community ceiling on this puzzle is 470; the full solution is 480.

Method	Best	Forced moves	Hardness peak	Area law	Rigidity	New basin
PALIMPSEST · from the corpus	463/480				●	no
PRIOR · from scratch	460/480		●		●	no
KEYRING · from scratch	460/480				●	new family
REPLAY · decode & replay	460/480				●	no
GAUNTLET · from scratch	458/480				●	new family
CLOISTER · anchor & constrain	453/480			●	●	no
MIDDEN · anchor & constrain	452/480			●	●	no
LADDER · concentrate effort	451/480	●	●			new family
LODESTONE · from scratch	451/480	●				no
MOSAIC · solve a piece exactly	448/480	●	●			no
BANDSAW · solve a piece exactly	437/480	●	●			no
STAGED · from scratch	436/480	●		●		no

How to read it#

Almost every method ends up against rigidity, the wall that says the great boards are isolated islands. The build-from-scratch and corpus methods (PRIOR, KEYRING, PALIMPSEST, GAUNTLET) try to reach a new island by steering construction with learned signal; they top out at 458–463 and a couple of them do reach genuinely new families, but none crosses to the ceiling. The concentrate and exact methods (LADDER, BANDSAW) instead attack the search itself (the high branching factor and the unsearchable peak) and pay for it at the endgame. The anchor methods (CLOISTER, MIDDEN) localise the damage but hit the area-law wall in the interior. No single wall is the whole story, and no method gets through all four.

Where each one stopped, and why#

The ceiling is never arbitrary. For each method, its own write-up records the exact reason the score stopped climbing, quoted here in one line.

PALIMPSEST (463/480). Reached 463, the project best, by reading the corpus to find which shared choices are traps and steering a 15-basin sweep around them. Forcing the search to avoid the traps directly made boards worse: the value was in where to look, not a hard rule.
PRIOR (460/480). Plateaus at 460: the learned position prior gets a from-scratch build into the 460 class fast, but the corpus signal alone is not enough to leave it.
KEYRING (460/480). Three learned signals voting (position, adjacency, 2×2 patch) reached 460 in a corner arrangement no board had cracked before, a new family, but the patch signal is marginal and the polish still caps at 460.
REPLAY (460/480). Replays the community's strict-460 boards exactly and reveals the move ordinary search misses: 4–5 cells that take two mismatches at once, unreachable for a search that allows at most one.
GAUNTLET (458/480). Running the beam in nine scan directions opened a brand-new 458 family (scan order is a stronger diversity axis than the random seed), but a second round topped out at 457 with no 461: the new family saturates like the others.
CLOISTER (453/480). As a standalone interior solver it confirms a real rim-compatibility bonus, but that bonus cannot be retrofitted after the fact (the same rigidity as the full board), so it settles in the low 450s.
MIDDEN (452/480). Choosing where (not when) the board may break extends the perfect run from 153 to 167–174 cells, but the dispersed geometry still fails at the endgame: nothing absorbs the last damage.
LADDER (451/480). Floods cheap probes and promotes the deepest, reaching a 451 strict board with no record to copy, the first escape from the universal 444–450 band, but the supply of perfect openings runs out and the rungs all converge to one ceiling.
LODESTONE (451/480). A scarce-demand prior used only as a tiebreaker lifts the from-scratch median by two (449→451) and tightens the variance, but any larger weight collapses it: scarcity is a real but weak signal, and it never touches the basin ceiling.
MOSAIC (448/480). Composes exact 4×4 block solutions with soft seams, reaching 448 from scratch, but the shortfall lands almost entirely in the last three corner blocks, where the piece pool runs thin: the same piece-theft, now a single bright spot.
BANDSAW (437/480). Solves an endgame band to proven optimality, and in doing so measures the exactness wall: the search tree grows about twentyfold per extra allowed mismatch, on both sides, so meeting in the middle stops paying at full size.
STAGED (436/480). Builds the whole board with no pre-set frame and an emergent border, reaching 436, well below the records. That gap is the finding: it measures exactly how much the usual frame-first anchor is worth.

The shape of the gap#

Read down the table and the lesson of the whole project is visible at a glance: the methods that move the score change the shape of the search, whether a scan order, a learned prior, or a confined region, never its raw speed. And every one of them stops at a wall that is global, not local. The ten edges from 470 to 480 are not a polishing problem; they are on the far side of all four walls at once.

A separate 2026 campaign by William Millilaw reached the same conclusion from the diversity side. Sweeping the full method list, he found that almost everything (adaptive local search from scratch, snake placement, the top of a trained generator's distribution) keeps rediscovering the same handful of basins, and only parallel tempering reliably produced genuinely new ones, boards far apart from the known set. Even that plateaus quickly. His reading is the one this table keeps making: the bottleneck is not the score a method reaches but the number of distinct basins it can find, and no method in the standard arsenal finds enough of them.

Each saturation line is distilled from the project's lab notebook (one entry per experiment). The four walls are corroborated by the published literature: the 17-colour phase transition by Ansótegui, Béjar, Fernández & Mateu, "How Hard is a Commercial Puzzle: the Eternity II Challenge"; the missing gradient and deep basins by the Eternity II local-search literature.

Which wall stops which method

What each wall rests on#

The four walls, in one line each#

The map#

How to read it#

Where each one stopped, and why#

The shape of the gap#

Keep exploring

Referenced by

What each wall rests on#

The four walls, in one line each#

The map#

How to read it#

Where each one stopped, and why#

The shape of the gap#