Five of the 22 edge colors never appear on any of the 196 interior pieces. In
the standard numbering they are colors 1 to 5: the frame colors. Each occurs
on exactly 24 half-edges, all of them on the 60 border pieces, for a total
supply of 120. Ring purity is the statement that the piece data leaves this
supply no choice at all: every valid solution spends the entire 120 on the
120 ring-facing edge slots of the border ring, with zero slack. The border is
a self-contained sub-puzzle, and its only contact with the interior runs
through 56 inward-facing edges.
120 = 120
ring demand vs frame supply, zero slack
5 × 24
frame colors × half-edges each
56
inward edges coupling ring to interior
The proof rests on three properties you can verify by scanning the 256 pieces
once. No search, no sampling.
First, frame colors occur only on border pieces. The checker does not trust
the labels: it derives the frame set from the data, as the colors with zero
occurrences on interior pieces, and finds exactly five.
Second, every one of the 56 edge pieces carries exactly two frame-colored
slots and one non-frame slot among its three non-zero slots, and the
non-frame slot sits opposite the gray slot. This holds 56 out of 56 times.
Third, both non-zero slots of every corner piece are frame-colored, and the
two gray slots are adjacent. This holds 4 out of 4 times.
For the record, the rest of the palette is as flat as the frame part: of the
17 interior colors, 5 occur 48 times and 12 occur 50 times. Brendan Owen
measured exactly these counts the week he digitised his set
(msg 1054), and the strict
separation of the two palettes was a design decision
(msg 1947); the
design-recipe page tells that story.
An edge piece sits on the rim with its gray slot facing outward, so the slot
opposite the gray one faces the interior. Its inward neighbour is an interior
piece, which by the first fact has no frame color to offer. The inward slot
therefore cannot be frame-colored. By the second fact the edge piece owns
exactly one non-frame slot, and it sits exactly opposite the gray slot, which
is the inward position. So the two frame slots are forced onto the two
ring-facing positions, along the rim. Corners contribute both their non-zero
slots to the ring by the third fact.
Now count. Demand: 56×2+4×2=120 ring-facing slots.
Supply: 5×24=120 frame half-edges. The two numbers match exactly.
Every frame half-edge is consumed on the ring, none is left over, and no
non-frame color ever appears on a ring joint. That is the zero-slack part:
the frame-color budget is spent to the last half-edge.
The saturation has a clean graph-theoretic reading. Take a multigraph on 5
vertices, one per frame color, and one edge per border piece, joining its two
ring-facing colors. A valid arrangement of the border ring is exactly an
Eulerian circuit of this multigraph: walking the ring reads off a closed
trail that uses every piece-edge once, and conversely.
On the real instance the multigraph has 60 edges, every degree equal to 24,
and it is connected, so an Eulerian circuit exists, as it must, since a full
solution exists. Of the 60 edges, 14 are self-loops (pieces showing the same
frame color on both ring slots), and the edges span all 15 possible color
pairs, the 10 unordered pairs plus the 5 loops.
Two strength measures come with the theorem, one exact and one sampled.
The exact one is first-step branching. Fix any placed border piece; the next
piece along the ring must match the exposed frame color. Of the border
pieces, only those incident to that color qualify: depending on the color,
20, 21, 21, 22 or 22 of them, mean 21.2, against 59 candidates for an
unconstrained ordering. Color matching alone cuts the first branching factor
by 2.78x.
The sampled one is a sequential importance sampling estimate over 1000 seeded
greedy ring builds. The mean log10 path probability of a completed build is
-27.30 (sd 0.64) on seed 1 and -27.39 (sd 0.62) on seed 2, against a uniform
baseline of log10(1/59!)=−80.14. Read that as: the color-matching law
concentrates probability mass by roughly 53 orders of magnitude relative to a
uniform ordering of the border pieces, and still leaves a residual rarity
near 10−27. A purely greedy color-matching walk accordingly almost never
finishes a ring: 8.7% completion on seed 1, 9.3% (seed 2, run for batch consistency; the committed JSON records seed 1) on seed 2, with the source
run at 9.4%. All 87 completed seed-1 builds closed into circuits, open ends
matching, which is the Eulerian structure showing up in the sample.
The exact count of undirected Eulerian circuits of the real ring multigraph
is open. A single-orientation shortcut through the BEST theorem was attempted
during the original investigation, caught as an undercount by a K5 sanity
check, and withdrawn; no circuit count is claimed anywhere. The greedy
completion measure also ignores the geometric corner/edge alternation of the
physical ring: it measures the color-matching law in isolation, not the full
ring constraint.
The theorem splits the puzzle along the rim. Inside the ring, the 56 edge
pieces each show one color from the 17-color interior palette, and those 56
inward edges are the border's entire interface with the 196-piece interior;
the bookkeeping across that seam is the subject of the
border-balance page. The visible face of the
same split, the five rare colors quarantined at the rim, is on the
rare-color-geography page. And this
page is the detailed account of one law in the
theorem sweep, alongside the instance's other
exact structure.
Every number above is recomputed by the committed checker in the
reproduction topic linked under sources: a single Rust program that loads the
official instance, derives the frame colors from the data, verifies each
clause of the theorem exhaustively, builds the ring multigraph, runs the two
seeded Monte Carlo measures, and emits one JSON file (committed as
results/ring_purity.json) holding every figure quoted here. The
deterministic clauses reproduce byte for byte; the sampled ones agree within
sampling error across independent seeds.