# Results 0.1 — geometry helps, one harmony coordinate does not suffice

The frozen test uses NIST average phase-change values for the three C5H12
isomers.  Their computed carbon-skeleton Wiener indices are 20 for n-pentane,
18 for isopentane, and 16 for neopentane.

Training on the two structural endpoints gives

\[
\widehat T_b(W)=6.65W+176.2\ \mathrm{K}.
\]

The single held-out prediction is therefore 295.9 K for isopentane.  NIST's
average is 301.1 K, so the absolute error is 5.2 K.  At 293.15 K the prediction
places isopentane on the condensed side of its boiling boundary; its measured
fusion point is 113 K, so the observed equilibrium phase at about one
atmosphere is liquid.

This is better than composition alone in an exact, limited sense.  Any model
that maps the common formula C5H12 to one common boiling temperature has
worst-case error at least half the observed 26.6 K range, or 13.3 K.

The result does not generalize to all phase boundaries.  Fusion temperatures
are 143.4 K, 113 K, and 255 K: they are not monotone in the same geometry
coordinate.  Solid packing and symmetry add information absent from the
one-number descriptor.

The elemental control is also negative.  Silver and palladium share the frozen
coarse signature period-5/fcc/d10, yet Matula's recommended 300 K total
resistivities are 1.629 and 10.54 in units of 10^-8 ohm m, a 6.470 ratio.  Any
one-value predictor for that shared signature has worst multiplicative error at
least sqrt(6.470) = 2.544.  Valence occupancy, band structure, Fermi surface,
phonons, defects, and scattering lifetimes cannot be replaced by the coarse
label.

All 12 unit tests and the separately implemented Floyd-Warshall/scalar audit
pass.  The audit checks three graphs, three phase classifications, two bounds,
the held-out prediction, and the element ratio.