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.
Reproducedeterministic — reproduces byte-for-byte·reruns the search·Budget: well under a minute for the 500-frame main run on one core (; a preflight at the default scale completed in about 7 s)
Hardware & run
Native runCPU only
0.016core-hours
Cores
8
RAM
16 GiB
GPUs
0
CPU
Apple M1
Machine
MacBook (Apple M1, 8 cores)
Budget
well under a minute for the 500-frame main run on one core (; a preflight at the default scale completed in about 7 s)
Start
deterministic: seed0 1 for the main run, seed0 7001 for the robustness pass
Regenerates the 500-frame main run and the 100-frame robustness pass and reproduces results/manifold.json byte for byte; it re-measures the mechanism only, no completed boards are produced
Border-first pipelines such as
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.
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:
every legal same-class exchange and its BB cost, pooled over all frames;
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;
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;
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.
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.
▶Expected vs measured (main run, 500 frames)Explore →
Quantity
Expected (source)
Measured
Free exchanges per perfect frame
45, every frame
45 on all 500 frames
Exchanges that raise BB
0
0 over 773,000 pooled pairs
BB-cost shares (-4 / -3 / -2 / 0)
63.9 / 1.2 / 31.8 / 3.1 %
63.8 / 1.4 / 31.9 / 2.9 %
BB along a 100-swap free walk
60 at every checkpoint
60 at every checkpoint
Free moves available along the walk
45 at every checkpoint
45 at every checkpoint
Ring slots changed after 100 free swaps
36 of 60
33 of 60
Free swaps preserving inward colour
0 of 45
0 of 45
Distinct rim-target vectors visited
292 (longer source walk)
99 of a possible 101
Dead frames (any unfillable cell)
180/500 (36.0%)
180/500 (36.0%)
Dead at the interior scan-start cell
55/500 (11.0%)
46/500 (9.2%)
Dead-cell histogram (1 / 2 / 3 cells)
155 / 24 / 1
154 / 25 / 1
Dead frames revived by free swaps only
100%
60/60, BB still 60 on every one
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.
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.