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 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.
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 1015 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.
On the official 256-piece set with the five official clues pinned:
- 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);
- 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;
- 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.
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.
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.
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 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.
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.