Here is the intuition almost everyone starts with: get close, then tidy up
the last few mismatches. Swap a couple of pieces, rotate a few, and surely
the score creeps up to 480.
It doesn't. On every top board we have tested, the last mismatches are
locked. Move anything nearby and the score stays the same or drops. The good
boards aren't almost-solutions waiting for a polish; they're isolated points
with nowhere better to go next door.
Before any solver gets involved, the cheapest checks can be done by brute
force: try every small move there is, and count.
Here is a pocket-sized version you can hold in your head. The community 469
board and a known 466 board differ in exactly four cells: the same four
pieces, cycled one position along a bottom-row strip. Enumerate all 6,144
arrangements of those four pieces in those four cells (24 permutations times
256 rotation combinations) and the maximum is 469, achieved uniquely by the
incumbent's own arrangement. The 466 board is one of the 6,143 worse ones.
Two good boards can sit four cells apart with no path between them, or above
them, through those cells.
Scale it up. On one of our 459 boards (strict five-clue scoring; our numbers
sit below the community bests, see the records page for
context), every non-identity rotation of every piece was tested: 768 of them,
zero improvements; every real rotation strictly hurts. All 32,640 pairwise
piece swaps: zero improvements. Swap-plus-rotation combinations: 47,164 of
them, drawn from a 5,000-pair sample of the 522,240 possible, zero again;
that one is a large sample, not a full enumeration. There is also an
analytical reason rotations can never even be neutral: a 180-degree rotation
leaves the local match count unchanged in every possible neighborhood only if
the piece's top and bottom colors match and its left and right colors match,
and zero of the 256 pieces have that symmetry (none is fully
rotation-symmetric either). No rotation is ever free. The same cheap probe
run across eleven of our high boards, scores 454 to 459, gives 8,448
single-rotation tests and not one improvement.
On a 457-scoring board of ours (matched-edges convention), the enumeration
was pushed to every move touching up to five pieces: 1,024 single rotation
flips, 6,720 adjacent-pair rotations, 20,240 non-adjacent transpositions,
32,796 3-cycles on the 38 mismatch-adjacent cells, 982,200 4-cycles, and
28,480,440 5-cycles. Roughly 29.5 million moves, zero improvements, plus two
branch-and-bound sweeps (a 38-cell permutation search and an 82-cell
radius-1 halo) that also found nothing. That same 457 board was reached 11
independent times by a hot-restart search, byte-identical every time: the
search does not orbit near a good board, it collapses onto one exact point
and stays there.
Even the moves engineered to look free fail. Some piece pairs share three of
their four edge colors, so swapping them looks almost neutral. On the
community 469, all 114 such single swaps were tried: exactly one keeps the
score at 469 (producing the only known sibling board), 2 give 468, and 111
give 467. All 6,441 double swaps: none reaches 470, the best gives 468.
Across 6,555 perturbations built specifically to be nearly score-neutral, the
469-score level set contains exactly two boards.
The numbers explain themselves. At score 459 a cell contributes about 3.6
matched edges on average, while a random piece dropped into a fixed
neighborhood expects only about 0.18 (roughly four edges times one chance in
22 colors), so a random swap expects to lose about 6.8 edges. A loose
probabilistic bound caps the chance of an improving swap at about 5 percent;
the measured frequency is below 3×10−5. Each placement in a top
board is co-adapted to its neighbors far beyond what chance alone would
predict. One scope note: each census here is exhaustive within its move
class but runs on a single board (the rotation and swap censuses on one 459,
the five-piece enumeration on one 457, the near-twin census on the 469); the
exact proofs below are what generalize them.
The result does not rest on trying a lot of swaps and giving up. It comes from
an exact question put to an integer-programming solver: take a region of the
board, free every piece inside it, and find the single best way to fill it back
in. The solver searches that whole local space exactly, not by sampling.
The answer keeps coming back the same: the best arrangement is the one
already there. We proved it for region after region, on board after board,
and out to a radius of four cells; the largest single region closed to a
certificate is 79 cells, nearly a third of the board. Every time, no
improvement exists.
One caution learned along the way: a cold exact re-solve of a large region
can silently under-perform. Asked to rebuild the last three rows of a strong
board from scratch (48 cells), the solver could not even rediscover the
board's own tail in the time given, reaching 68 matched edges against the
incumbent's 72 after 201 s (a single run). At larger windows, "found nothing
better" can be plain solver difficulty rather than proof. A rigid verdict is
only trusted here when the solver starts from the incumbent, so anything
better is strictly easier to find, and it still certifies a zero gap.
The exact question, stated precisely. Around each defect cell take the
halo-r region: every cell within r steps. Free all pieces inside it,
keep the rest of the board fixed as a boundary, and solve an integer program
for the best legal refill. Binary variables xc,p,θ∈{0,1} place
piece p at rotation θ in cell c; one-piece-per-cell and
one-cell-per-piece constraints make it an assignment problem; the objective
maximizes matched edges, counting the fixed boundary. A 64-cell region is about
18,700 binary variables. Branch and bound then either finds a strictly better
refill or proves, with a dual bound, that none exists. The definition worth
keeping: a board is locally rigid at radius r if, for every connected
component of its mismatch-adjacent cells, freeing all cells within distance
r of it and solving exactly yields no improvement.
It never finds one. The proofs, region by region. These MIP runs come from the
project's research notebook (the 2026-05-16 paper linked in the sources); the
table below is not yet re-run under this site's committed-results pipeline, so
its solver times and dual bounds are not independently checkable from this repo
the way the SAT-halo result further down is. The same provenance note covers
every census, enumeration and SAT escalation added to this page; the radius-4
SAT reproduction is the one result committed in this repo:
Region
Board
Cells
Solver time
Result
halo-2, per-component (×8)
3 basins (469, 459, 458)
2–34
seconds each
all Δ = +0, proven
halo-1 joint
McGavin 469
37
895 s
Δ = +0, proven
halo-1 joint
Local 459
59
1800 s
Δ = +0, proven
halo-1 joint
three separate 458s
57, 62, 64
180 s each
Δ = +0, proven
halo-1 per-component
a second, disjoint 459
52 + 4
~60 s
Δ = +0, proven
top-3 rows (a band, not a halo)
McGavin 469
48
70 s
Δ = +0, proven
halo-3, per-component
McGavin 469
42 + 47
929 s
Δ = +0, proven
halo-3, densest cluster
a 458
79 + 19
1200 s + 0.2 s
Δ = +0, proven
halo-4, component 0
McGavin 469
57
1200 s
Δ = +0, proven
Across at least six distinct boards and more than twenty regions the answer
is invariant: the incumbent board is locally MIP-optimal, out to halo-4 for
McGavin's 469. Two of the rows deserve a word. The top-3-rows row is a full
geometric band, not a halo: it frees clean cells alongside broken ones, so it
additionally rules out moves that trade pieces between broken and clean cells
inside the band. And the 79-cell region, the largest ever closed here, sits
on a 458 rather than the 469 because solvability tracked defect density,
not raw region size: dense-defect regions close while larger sparse ones time
out.
The one region left with a gap, the top-four rows, 64 cells, still gives a
sound result: a dual bound of 123 against the incumbent's 116, so even the
unclosed case cannot exceed the wall by much (it implies a board-wide bound
of ≤ 476).
Not every solve finished, and the ledger keeps those too. The 54-cell halo-2
joint solve on the 469 sat at its root relaxation for 55 minutes and was
killed; the top-5-rows band (80 cells) timed out; on the three 458s the two
big halo-2 components hit a 120 s limit reporting +0 without a certificate,
which counts as "no improvement found", not as proven. One more unfinished
row widens the evidence to a different scoring convention: a 460 board played
without the clue constraint was given a halo-1 joint solve (42 free cells,
full piece freedom) and a 900-second run found no improvement but left a 9.5
percent dual gap open, so that row too is "nothing found within budget", not
"proven optimal". Single board, single run.
Nor is this one unlucky board. A second 459-scoring board, found by a
completely different construction path, agrees with the first on only 3 of
256 cells, essentially just the clue cells; the two boards are structurally
disjoint, and the second is also proven rigid at halo-1 (two components, 52
and 4 cells, +0 proven in about a minute). At halo-2 its small component (7
cells) proved +0 in 18 s; its large component (67 cells) was inconclusive
after about 13 minutes and stays open. The conjecture this supports, stated
as a conjecture: the 459 level set is a disjoint union of many rigid points,
and climbing past it needs a board-spanning rearrangement or a different
start, never a local fix.
The badge says proven rather than conjectured for the regions actually
closed; the general statement over all boards remains a conjecture, supported
by every region tested to date.
The MIP frees a region and refills it from the whole piece set. There is a
complementary move class it does not cover: pick k cells and try every way
to permute and re-rotate the pieces already sitting on them, with the rest
of the board frozen. A different move algebra, and it gives the same answer.
On a 452 board of ours (strict five-clue scoring, 2026-07): k=1,
exhaustive over all 251 free cells, zero improving moves. k=2, exhaustive
over all 31,375 non-clue pairs, zero. k=3 to 6, about 29,000 sampled
clusters (near-defect, whole-board, connected and disconnected), zero; that
tier is a large sample, not an enumeration. For k=7 and 8 the sampling
was upgraded to a genuine enumeration: every connected 7- and 8-cell subset
of the defect-touching cells on two independent boards, 849 subsets in all
(231 on the 452, 618 on a structurally unrelated 451 that differs from it at
249 of 256 cells), all solved to completion, zero improving, zero timeouts;
plus all 7,845 connected 7-cell subsets within one step of a defect edge,
zero again. Roughly 60,000 evaluated cluster moves, and the best delta found
anywhere is exactly zero.
The zero is not a broken detector. As a positive control, a deliberate
two-piece sabotage (452 down to 447) was repaired by the same machinery in a
single 4-cell move, straight back to 452. And greedy local search over this
move class, started from three independently sourced boards (the 452, a
same-score twin differing at exactly 2 cells, and the unrelated 451), hit an
immediate fixed point in all 20 seeded runs: minimum, median and maximum
identical, zero escapes.
The detail that makes it sting: the 452 itself was born from exactly this
move class. A two-piece co-rotation lifted a 451 that was rigid under every
single-piece move. One genuine two-body correlated move existed; after taking
it, no correlated move of size up to 8 remains anywhere near the defects. The
rigidity radius grows as the board improves.
Changing the geometry of the freed set does not help either. Free contiguous
row strips instead of blobs: 54 of 54 single-row strips through the two
boards' defect bands (widths 8 to 14, up to 15 cells freed) certify at
exactly the board's own score in under a second each, and 18 of 18 two-row
strips outside the worst band certify rigid up to 26 cells. In each board's
densest defect band the wider strips (16 to 26 cells) get hard to certify:
after escalating to 300 s and 3 seeds, 6 of 10 remain open with solver-bound
gaps of 2 to 6 edges. But across all 30 escalation runs the solver never
found any arrangement better than the board's own. That is a certification
gap, not evidence of an improvement, and the open strips are recorded as
open, not proven.
Rectangular windows tell the same story. On the 452, whose 28 broken edges
split into 22 concentrated in the bottom band and 6 scattered, every window
of up to 18 cells around every defect cluster (four windows, 9 to 18 cells)
re-solves to a certified optimum at exactly the board's own value. Each
defect is forced by pieces committed elsewhere; no window that merely
contains the defect can fix it.
Even letting pieces trade across the good/bad boundary moves nothing. On a
459 board of ours (strict five-clue), the bottom two rows (32 cells) certify
optimal in 49 s. A 46-cell window, and 52- and 72-cell windows that
additionally free 6 to 10 donor cells inside the perfect region, so pieces
can be exchanged between the clean and broken parts, all return the identical
board: zero
cells moved, not even a lateral equal-score shuffle (150 to 180 s each,
single seed, so these larger windows count as "no improvement and no movement
found", not as certificates). The same operator applied to a weaker 455 board
lifts it by 2 to 457, so it demonstrably improves boards not yet at their
fixed point; the 459 already is one. Inside every window the linear
relaxation believes about 17 more matches are available; the
each-piece-used-once constraint forbids them all. The binding constraint is
which pieces the rest of the board leaves over, never window size. The
hardness is global distinctness, not local matching.
Halos and windows are blobs. Try a board-spanning geometry instead: free an
entire 16-cell row, restrict it to pieces not used elsewhere on the board,
and enumerate alternative fills by dynamic programming. On a 460-scoring
board (mismatches all in rows 11 to 14) and on the community 469 (mismatches
in rows 0 to 4), every row's current fill is the best available, and several
rows admit no legal alternative at all: the placed chain is the unique way
to thread that row through the rest of the board.
The numbers. On the 460: rows 1 to 10 are perfect and admit exactly one
chain each, the placed one; rows 12 and 14 have zero alternative chains; row
13 has 176 alternatives, all worse, the best of them 10 edges down. On the
469: row 3 has zero alternatives; rows 2 and 4 have best alternatives 20
edges down; rows 5 to 14 are perfect, with the original among up to about
17,857 legal chains and none better. Jointly freeing two rows (32 cells,
internal seam unconstrained): the pairs (11,12), (12,13) and (13,14) on the
460 yield 0 to 4 alternative chains in total, all worse (12 to 16 edges
down) or infeasible. Three rows jointly (48 cells): rows (11,12,13) and
(12,13,14) yield zero alternative chains. These rows are not just optimal;
they are often forced. Uniqueness, not merely optimality. Scope: the
enumeration is beam dynamic programming (beam width 100,000 for one row,
5,000 to 50,000 for two and three rows), exhaustive in practice but not
certified the way the MIP rows are; call it beam-exhaustive. Two boards.
An exact companion on bands: strip the bottom 2 rows (32 cells) of a
460-scoring board entirely and let an exact constraint search enumerate every
legal refill from the freed pieces. Exactly 32 alternative fills exist, all
scoring at or below 460. Strip the bottom 4 rows (64 cells): the only
completion the exact search reaches is the original board itself. One board,
these two band sizes only; but on every band that could be closed exactly,
the current score is the true ceiling.
Where the last mismatches live, and why they're stuck#
On a 458 board of ours, the anatomy of failure is remarkably concentrated.
The border ring is perfect (60 of 60), the top nine interior rows are
perfect, and all 22 mismatches (4 border-interior, 18 interior-interior)
live in the bottom five rows, grouped into 10 tiny disjoint clusters
averaging 2.8 cells, the largest just 5. Each cluster looks trivially
fixable. Exact solves say otherwise: per-cluster refills, delta zero; halo-1
and halo-2 enlargements up to 16 cells, delta zero; and the union of all 28
defect-touching cells, solved jointly to proven optimality in 1.74 s, delta
zero.
This is the mechanism behind the whole wall in miniature: the defect region
is at its exact optimum given the pieces it was left with. The perfect top
of the board has consumed a specific piece set, and that commitment is what
caps the bottom. Fixing the last mismatches would require un-committing
pieces from the already perfect region, a large cross-region move, not a
local repair. (A linear relaxation reads a ceiling of 478 for this border;
that is a relaxation ceiling for this particular border, not a claim about
the true optimum.) One board, but the pattern, defects packed into a few
tiny clusters that exact refills cannot fix, is exactly what the halo table
above keeps finding on the other basins.
Two facts coexist on the 452 board (strict five-clue), and the tension
between them is the current state of the art. Fact A: no local move of any
tested kind beats it, and four independent framings agree (cluster moves,
window re-solves, an exact zero-gap certificate that the pool of pieces on
its defect-touching cells is placed optimally, and a thermodynamic probe that
either freezes at the incumbent or melts to random, with no gentle escape in
between). Fact B: every attempt to bound a large region from above fails;
the bounds are loose and do not converge. On the board's three-row tail, 8
seeds at 20 minutes each all returned exactly the incumbent, no seed ever
found anything better, while the solver's upper bound rose from 79 (at
300 s) to 89 (at 1200 s), moving away from the incumbent's 72, not toward
it. An interior linear-programming bound reads 478.5 out of 480: vacuous.
There is a cautionary tale inside that. An early reading of the 300 s run
was "a gap of 7, headroom to a higher score". The 8-seed chase showed the
gap was an artifact of a non-converging relaxation, not evidence of a
reachable better board. The reason the bounds stay loose: the scarcity that
makes this puzzle hard is second-order, a matter of which pairs of colors
are jointly scarce, and first-order matching relaxations, which is what LP
and propagation bounds are built on, cannot see it, no matter how long they
run.
So local rigidity (proven) and the open global gap are not in tension. The
incumbent is a deep local optimum, and a disconnected better arrangement
may exist that no local move reaches; today no bound rules it out and no
search finds it. Closing either direction of one large-region gap would be a
first: a better filling would be a new best board, and a bound driven down to
the incumbent would be the first proven local score ceiling. The
top-four-rows dual bound above (123 against 116) is the closest thing to
progress from above, and it is still 7 edges loose. Scope: Fact A is proven
on one board through multiple lenses; Fact B carries an 8-seed spread on its
decisive run.
Rigidity survives even when the move set is "borrow from every good board we
have ever found". Build an optimization in which each cell may take the piece
and rotation that any of 25 to 30 distinct high-scoring boards places
there, with each-piece-used-once enforced, and free the entire board, all
256 cells. The optimum is exactly the best board already in the corpus,
never a blend that beats it: 457 with all five clues pinned (strict
convention; 25-board corpus, solved in 24 s), and 459 without clue
enforcement (matched-edges convention). Region versions at growing radii, 77,
127, 191 and 252 free cells, are all delta zero in seconds to about 30 s.
These optima are certificates, per corpus; the corpus is from 2026-05, when
our bests were 457 strict and 459 matched-edges, both below the community
figures then and now (records page).
The corpus is not choice-starved. Every one of the 256 cells has at least 2
distinct options across the boards, and a typical piece appears at 6 to 19
different positions across the corpus. The diversity is rich;
piece-uniqueness fragments it. Boards from different corner-arrangement
families cannot be mixed at all, and adding the community 469 board to the
corpus makes the optimizer simply pick that board wholesale, at 469, rather
than blend it with anything.
The reading: the gap above the best known boards is a discovery gap, not a
recombination gap. The improvement is not hiding in any combination of what
is already known; it requires boards outside the entire known corpus.
This reframes the whole gap to 480. The distance from the best known board to
a solution is not a pile of small fixes waiting to be found. If it were, this
kind of local search would have found them. The barrier is that the good
boards sit at the bottom of their own little valleys, and the valley walls
are exact, not approximate.
It also tells you what won't work. Polishing, hill-climbing, and most
local-repair heuristics are trying to walk uphill from a frozen point. There
is no uphill. Reaching a solution needs a move that rearranges a large region
all at once, or a different starting point entirely, not a better polish. The
halo-1 joint proofs make that quantitative: any destroy-and-repair operator
whose reach is at most the halo, window sizes up to about 30 cells around the
defects, is mathematically exhausted on these boards. The freeze is not "our
heuristic is weak"; that entire operator class is finished.
Even the largest single move we know how to make does not escape. On our 461
board (strict five-clue; for how that number sits against the community
records, see the records page), the indecomposable
whole-cycle piece moves described on the
sigma-cycles page were applied atomically,
each followed by a local window re-solve to repair the disturbed cells.
Roughly 4,000 such moves: zero escapes, and the cheapest non-local exit still
loses at least a point (best result 460, two cells different). This rules out
only the local-cleanup variant; a globally compensating re-solve was not
tested. But a large move is not enough if its cleanup is local.
One more thing rigidity does not mean: hard to rebuild. Pin the first
fourteen rows of the community 469 and let a repair search rebuild the rest:
3 of 4 seeds get back to 469 (the fourth sticks at 454). Do the same with
our own record boards' prefixes: no seed ever exceeds 460. A small probe (4
seeds, 30-second repairs per board), but the shape is clear. The wall is not
about how hard a board is to reconstruct; it is about which valley the
board's skeleton belongs to. A prefix either admits a great completion or
it does not, and no amount of seed luck changes which.
Pin the top N rows of the community 469 board and let local search
complete the rest. The completion score is not gradual in N; it is a cliff
with decoys: N=1 gives 400, N=2 gives 382, N=4 gives 401, N=8 gives
418, N=12 gives 450, N=13 gives 455, and N=14 gives 469, an exact
reconstruction, zero cells different, with N=15 the same. At 13 pinned
rows there exists an alternative completion of the last three rows, using
the same pieces differently, that scores only 455 and is itself a trap:
local color matching admits multiple completions and the search cannot tell
which one extends. Getting 87 percent of a record right is not "almost
there". These are single completion runs per point, on minutes-scale
budgets, with no seed spread; read the threshold, not the individual scores.
Two side notes complete the picture. The sweep proves the completion
machinery is capable of reaching 469; the wall is in finding the first
roughly 224 pieces of the structure, not in any weakness of the polish step.
And influence does not flow the other way: a top row alone, one of at least
5×108 legal top rows, determines almost nothing. Four operator
presets of our local-search family, grown from a pinned top row, all stall
between 378 and 400; other search families are untested, so that negative is
scoped to our searcher. Meanwhile 363 of our own 455-plus boards used only
47 distinct top rows: the good boards cluster at the top long before the
bottom is decided.
Someone hit the same wall from the annealing side#
The MIP proof approaches the wall analytically. In June 2026 another researcher
ran headlong into it empirically. Working the strict five-clue board with a GPU
simulated-annealing solver (4096 parallel replicas), benj39100 reported two
things that read like a restatement of this page. First, the best score a run
could reach increased with distance from the current best board: to climb
from 429 toward 432 the winning perturbations had to migrate steadily further
out, because near the incumbent there was no improving move to find. Second,
across all their best boards the same roughly forty-two broken edges formed a
frozen "hard core" that local search never touched, so they had to add an
explicit term that rewarded prying that core open. A locally frozen incumbent
with no nearby gradient, and a small locked set of defects nothing local will
move: that is the rigidity wall, seen from a completely different method.
A third researcher reached the same conclusion with a third tool. William
Millilaw, working the ceiling independently, ran two tests. The first was a
freeze test: perturb the root of a top board, re-optimize, and see which cells
come back. On his best boards 93 to 100 percent of the cells returned to exactly
where they were, a frozen core the search could not move. The second was a
decision question for a SAT solver. Free the cells around the mismatches, demand
that every freed edge match, and ask whether any arrangement of those pieces
satisfies it. His solver answered UNSAT.
We reproduced that SAT test
on five public boards, with our own encoder and a positive control to make sure a
matched region comes back satisfiable. Across the four community record boards,
from Verhaard's 467 up to Blackwood's 470, and Riotte's 464 (the strict
five-clue record), every solve that finished is UNSAT out to a Chebyshev
radius-4 halo (this reproduction measures the free region by Chebyshev radius,
whereas the notebook SAT ladders further down count Manhattan steps, so the two
halo scales are not the same metric): no
local rearrangement of a record board's own pieces closes a single mismatch, even
when the freed region grows to well over a hundred cells. The radius-4 instances
are large enough that a few did not finish inside the solver's time cap; those
are recorded as timeouts and left open, not counted as proven, so the committed
table states exactly what was decided. Integer programming optimizes and bounds;
annealing runs into the wall by hand; SAT decides and returns a refutation. Three
methods, one answer.
A fourth tool agrees. A MaxSAT optimizer, a SAT variant that returns both the
best refill and a proof of its optimality, was pointed at sub-regions of an
earlier 454 board of ours: pin everything outside a region, free the region,
ask for the provably best refill. Every k×k window up to 5×5 was
proven optimal within its 60 s budget (the 6×6 windows exceeded the budget
and are recorded as not closed, not as proven), and the board's entire
45-cell defect zone, the region containing all 26 of its interior
mismatches, was proven optimal in 83 s: those 26 mismatches are the best
that region can do given the rest of the board. That 454 was itself reached
byte-identically from 4 different random seeds, the same
collapse-onto-a-single-point behavior as the 457 in the census above:
independent runs do not just reach the same score, they reach the exact same
board.
Push the SAT question further and the answer gets more dramatic. Take one of
our 459 boards (five clues enforced), free every cell within distance 10 of
its mismatches, 218 of the 256 cells free and only 38 pinned, and ask
whether any completion reaches a full 480. UNSAT, in 0.31 s. The ladder on
the way there: halos 0 through 5 all UNSAT in under 2 s each; halo 7, 173
free and 83 pinned, UNSAT in 0.17 s. The 38 cells still pinned at halo 10
are essentially the top two rows plus 3 of the 5 clues. So the top two rows
of this board are, by themselves, provably incompatible with any perfect
solution: any 480 must differ from this board somewhere inside those 38
cells. The rigidity is carried by a thin top-of-board skeleton, not by the
defect neighborhoods. At radius 15, only 4 cells pinned, the solver timed
out at 600 s; that case is open. One specific board; whether other
459-scoring boards share the same pinned-top rigidity is not known.
His earlier snake-and-local-search work put a number on why. The largest patch
any of his repair operators could rewrite in one move was about 48 cells, while
two of the known good boards, both near the ceiling, differ across roughly 225
cells. A move that can only touch 48 cells cannot cross a 225-cell gap, so no
sequence of them reaches a different basin. He looked for a chain of small
improving moves that would tunnel out and found none: zero improvements across
twenty-eight million four- and five-move combinations at the plateau. That is
the same wall stated as a budget. Local repair rewrites too little at once to
leave the valley, which is why escaping needs a move that rearranges a large
region all at once, exactly as the proof section above concludes.
Everything above asks whether a local rearrangement can close a mismatch.
There is a much bigger question: keeping only a board's border ring, does
any arrangement of the 196 remaining pieces reach a perfect 480? On our
459 board (matched-edges convention, canonical clues) the answer is a proof
of no, at every scale. Free the 32 mismatch cells: UNSAT in 0.09 s (14k
variables, 81k clauses). Free everything within 5 steps of a mismatch, 140
cells, more than half the board: UNSAT in 1.56 s. Free the entire
interior, all 191 non-clue interior cells, keeping only the 60-piece border
ring: UNSAT in 1.37 s (160k variables, 5.4 million clauses). The
intermediate rungs, halo-1 at 70 cells (0.33 s), halo-2 at 93 cells
(0.89 s), halo-3 at 109 cells (0.80 s), are all UNSAT too. The solver is
kissat, linked in the sources.
And it is not one board's bad luck. Nine distinct border-ring configurations
were tested, spanning four different corner arrangements: the community
469's border, our 459 and 458 record borders, a 435 board derived from
belief propagation, and five systematically enumerated border partials.
Every single one is UNSAT for 480, each in under 2 seconds. Most notably,
the community 469's border, 11 mismatches from perfection, provably cannot
host any 480 interior: UNSAT in under 0.01 s. The encoder was round-trip
validated on small solvable puzzles, where fresh solves decode to verified
perfect boards and pinning a correct border returns SAT, the same
positive-control discipline as the reproduction above; the UNSAT verdicts
are real, not artifacts.
The open edge of this result is its best summary. Only nine border
configurations exist in our corpus, and a 480-compatible border certainly
exists: the puzzle was constructed from a solution. It just is not any
border that any record-class search has ever produced. Polishing near a
record is not merely slow; under that record's own border, reaching 480 is
provably impossible. A solution requires a different border.
The page so far proves the wall; here is the closest thing we have to a
reason. Model the set of boards at a fixed score as a graph whose edges are
moves touching at most k cells, and ask for the smallest k that connects
it: the connectivity radius. On small edge-matching instances where every
board can be enumerated exactly, two laws emerge. First, the radius is small
across the bulk of the score spectrum and jumps to the whole board exactly
at the top score: on a 6-cell instance, k is 2 to 4 cells at interior
scores and 6, the entire board, at the maximum; on a 9-cell instance the
ratio k/N sits around 0.22 to 0.44 in the interior and 0.78 at the top.
Second, rigidity switches on with color richness: sweeping the palette on
9-cell instances, the mean minimal move support rises from 1.4 to 7.6 cells
as the colors go from 2 to 8, and the fraction of instances whose perfect
solutions need a full-board move to interconnect rises from 0 to 70 percent.
At toy scale there is a theorem, proved by a forcing argument and verified
by exhaustive enumeration: if the color design is rich enough that each
exposed edge color admits at most one legal tile during a fill, then two
distinct perfect solutions share no cell at all, so any move between them
must touch every cell. At color richness comparable to the real puzzle's,
every tested instance was rigid in exactly this sense: zero shared cells
across every pair of perfect solutions.
For Eternity II itself this is a conjecture, and we label it as one. With 22
interior colors, each reused roughly 24 to 50 times across 480 interior
adjacencies, the real puzzle sits deep in the rigid regime, so distinct
near-perfect boards should be near-orthogonal (which matches the observed
225-cell difference between good boards above), and no bounded-size move
should connect distinct optima. The "at most one tile" condition is not
strictly true on E2, so the supported claim is "a large fraction of the
board", not "provably all 256 cells".
The corollary is the satisfying part: local search freezes precisely at
high scores because that is where the move graph disconnects. The interior
of the score range is easy to walk, which is why reaching the 450s and 460s
is routine; the top is a set of isolated points, which is why polishing dies
there. One theory, both halves of this page's story. (The mathematics is the
Markov-basis and Graver-basis circle of ideas from algebraic statistics; see
the sources. The only cheap top-level moves the theory would allow are
global symmetries, and this piece set has essentially none.)
The next piece of the picture
If you can't improve a board locally, maybe you can hop from one good board
to another. That fails too, for a related reason: see why basin-hopping
is impossible.
The proofs use integer programming on regions of each board; they run for
minutes per region on a solver, so they aren't reproduced live here. The
boards themselves are bundled and checkable in the viewer.