The command reproduces exactly, in-repo: the horizontal bound lambdaH and the width-1 and width-2 entropy densities h(1), h(2) from the row-transfer eigenvalues; the interior block counts A(n) for n up to 4 and the distinct-piece counts B(n) for n up to 3 (B(2)=4059952 cross-checks the subgrid-placement-counts table); and the area-law exponent alpha near 0.044 fit over that range. The wider-strip points h(3), h(4) and the limit h_infinity near 0.67 are still carried from an offline sweep, and the rho(n) curve and the ~155-cell collapse point beyond n=3 are extrapolations of the in-repo fit, not exact counts.
Forget the use-each-piece-once rule for a moment and treat the 196 interior
pieces as reusable tiles. Count the valid all-matched patches and they grow
exponentially with size: there's no shortage of locally-valid ways to tile.
The matching grammar is rich, not restrictive.
We can measure that richness exactly. For a strip of width n, the growth
rate per cell is an entropy density h(n), and the sequence decreases toward
the true two-dimensional value as the strip widens.
Now put the use-each-piece-once rule back, and ask how often a color-valid
n×n patch actually uses distinct pieces. Call it ρ(n). We count
it exactly, in-repo: for each interior block size we count A(n), the
color-valid fillings when pieces may repeat, and B(n), those that use
distinct pieces, and take ρ(n)=B(n)/A(n). It collapses, and it
collapses in the area, not the perimeter:
ρ(n)≈exp(−αn2),α≈0.044.
The exponent is fit by least squares over the exactly-counted range: A and
B are computed here for n up to 3 (and A up to 4), with
B(2)=4,059,952 matching the in-repo sub-grid reference table. The
per-block exponent is still rising over that small range (0.029 at n=2,
0.048 at n=3), so 0.044 is a lower estimate of the large-patch value. At
n=4 the reusable count A(4) is already 6.3×1016, past exact
distinct enumeration in reasonable time, so the curve past n=3 is an
extrapolation of the fit.
An area-law decay is brutal because area grows quadratically. Extrapolating,
the fraction of realizable patches drops below one in a thousand near 155
cells. (An earlier off-site study reported a steeper α≈0.085 and
an ~80-cell collapse; the exact in-repo counts over n≤3 give the gentler
figures here, with the rising per-block exponent consistent with the off-site
value being reached at larger blocks.)
Loading chart…
Fraction of color-valid patches realizable with distinct pieces. The exponent is the in-repo least-squares fit over the exactly-counted blocks (n up to 3); the curve past that point is an extrapolation.
The same idea on the real pieces, exactly counted. Step the block size and
watch how many colour-valid blocks survive the use-each-piece-once rule.
▶Interactive: distinctness collapse and rho decayExplore →
Watch scarcity bite, block by block
Two exact counts on the real Eternity II pieces. Reusable: how many colour-valid n×n blocks exist if pieces may repeat, the richness of the matching rules. Distinct: how many survive the use-each-piece-once rule. Step n up and watch the gap open.
Block size:
colour-valid blocks (pieces may repeat)4.55×10^6
…that use distinct pieces4.06×10^6
89%
survive the distinct rule
freedom per cell (entropy density)1.67
Even with reusable pieces, each added cell adds less freedom than the last; the grammar alone tightens as blocks grow.
The distinct-realizable fraction collapses with the AREA of the patch, not its perimeter, and area grows as n². The in-repo fit over blocks up to 3×3 gives an area-law exponent near 0.044; extrapolating, the fraction drops below one in a thousand near 155 cells, the scale of the smallest moves between the best known boards. The hardness isn't in matching colours (that grammar stays rich); it's in the quiet once-each rule, whose cost compounds over area.
A collapse in the low hundreds of cells is on the scale of the smallest moves
that separate the best known boards. The matching grammar stays rich, then the
distinctness rule collapses it over the area. So the wall isn't in the part
that looks hard, matching colors; it's in the quiet rule that each piece is
used once, whose cost grows with area, on a board just large enough for it to
bite.
The per-width entropy has a well-defined limit. Joining an n1-wide and an
n2-wide strip side by side only adds a seam constraint, so the eigenvalues
satisfy
λn1+n2≤λn1λn2.
Taking logs makes logλn subadditive, and Fekete's lemma gives the
limit as an infimum, which is exactly why the curve above decreases:
h∞=n→∞limnlog10λn=ninfnlog10λn.
The upper bound is the purely horizontal rate: ignoring vertical constraints
only adds patches, so
0<h∞≤log10λH=1.6645,
with λH=46.18 the spectral radius of the horizontal
color-compatibility matrix. Positivity holds because the grammar supports
exponentially many chains, so the density is strictly between zero and
1.6645, measured near 0.67.
What the theorem proves is that a positive limit exists and is bounded above
by log10(46.18)=1.6645. Its specific value near 0.67 is carried from an
offline sweep. The area-law exponent α≈0.044 is fit here from the
exact in-repo block counts over n≤3; its extrapolation to larger patches
(and the ~155-cell collapse) goes past the range counted exactly. The theorem
does not fix those numbers.