A belief-propagation free energy computed over a board's last-row leftover pieces rank-predicts how well that row can still finish. The signal is not a raw-score proxy, survives a change of producer, and dies past one row.
Regenerates the 120 greedy tops, the exact last-row labels, and the full correlation table (qualitative tier: the source study's beam-top ensemble is not regenerated)
A constructive builder ranks its partial boards by matched edges so far. That
number says nothing about whether the leftover pieces can still finish the
remaining cells well. FROSTLINE asks whether a cheap physics-style quantity,
the Bethe free energy of a belief-propagation model over the last row's
leftover pieces, predicts the exact best score that row can still reach. At
one row it does: Spearman rho about -0.75, cross-validated, and the signal
survives a change of producer.
Freeze a full board's first fifteen rows and take the sixteen pieces the board
itself placed in row 15 as the leftover pool, which guarantees the last row
can be filled at all. Two numbers are then computed per board:
The label. An exact solver (a bitmask dynamic program over piece subsets
and seam colours, exact in milliseconds at 16 cells) finds the best
achievable tail score: the 15 edges inside the last row plus the 16 seam
edges against row 14, out of 31.
The predictor. A factor graph over the sixteen empty cells, with cell
states drawn from the leftover pool and pairwise factors that reward colour
agreement at inverse temperature beta. Sum-product belief propagation runs
on it and the Bethe free energy is read off. Lower free energy should mean a
more finishable tail.
Two modelling choices matter. The tail is a max-matching problem, not a hard
constraint problem: a colour mismatch costs one point instead of being
forbidden, so the graph is a finite-beta soft model rather than a
satisfiability instance. And the seam against the frozen rows must be a soft
factor; the source study measured that a hard seam filter contradicts itself
on 5 of 16 cells of real near-optimal tops. At one row the residual graph is a
chain, so belief propagation is exact there and the free energy is the true
log partition function.
The source study found the effect on 103 distinct tops from a
width-2048 beam producer, raw scores 446 to 452. The committed reproduction
regenerates everything in a different regime: 120 seeded greedy tops, raw
scores 361 to 396, best-tail labels 16 to 22. The expected column below cites
the source study; the measured column is this run.
▶Expected vs measured, last row (16 cells)Explore →
The pre-registered gate (|rho| at least 0.4, or AUC above 0.75 or below 0.25)
is cleared at every beta tried, with the source's sign. Two readings of the
table:
Not a raw-score proxy. Controlling for the board's own matched-edge
count, the partial correlation stays at -0.74. In the greedy regime the raw
score is nearly uninformative about the tail (+0.08 against the source's
+0.605), which makes this control stricter here, not weaker.
It measures the board, not the producer. The signal was found on beam
tops and reproduces on greedy tops, a different producer with a lower and
wider quality distribution. A statistic that survives that swap is reading
the leftover pieces, not an artifact of how the board was built.
Single boards make the point concrete: the reproduction's
worst-tail top scores 375 raw with a best achievable tail of
16, while its best-tail top scores 372 raw with a tail of 22.
The lower-raw board has the better ending.
The signal dies past one row. Holding the same label and cutting the
board higher, the source's correlation degrades from -0.74 at 16 residual
cells to -0.19 at 32 and -0.12 at 48, as the mean per-cell domain grows from
12.5 to 32.6 to 72.2. FROSTLINE is a last-row finishability oracle, not a
mid-construction steering signal.
It does not (yet) buy better boards. In the source study, using it to
select which partials get exact-solve compute does not beat selecting by raw
score: the realized full score is dominated by the raw-score spread, while
the free energy predicts only a small tail delta. The discriminator is real;
the selection value, in this form, is not.
Secondary quantities are regime-dependent. The Bethe entropy correlation
flips sign between regimes, and two quantities that were null on beam tops
carry weak signal on greedy tops. None of this touches the free-energy
headline, but all of it is recorded as caveats in the committed plan.
Nothing here moves a full-board score; for where the frontier actually
stands, see the records page.
The run is seeded and deterministic: just research-frostline regenerates the
120 tops, the exact labels, and the full correlation table in about two
minutes on one core, reproducing the committed frostline_r15.json. This is a
qualitative-tier reproduction: the effect's sign and strength band are
regenerated; the source study's beam-top ensemble is not. The full
expected-number list and the scale-faithfulness argument live in the committed
plan linked above.