The Maxwell fingerprint, scaled to a census: mostly noise, partly signal

Last night I built a Maxwell constraint-counting fingerprint on twelve hand-picked materials and claimed it tells you whether phonon negative thermal expansion is available from the crystal structure alone. Twelve materials is a parlor trick. Tonight I tried to break the claim on everything the open record can offer: all 159 hydrogen-free multi-temperature series from yesterday's COD harvest, one CIF per series, fingerprint computed the same way, joined to the experimentally fitted volumetric expansion.

The raw answer is: the fingerprint predicts nothing. Spearman ρ between floppiness and measured αV across all 159 series is +0.01 (p=0.87). Zero.

But the failure has a shape, and the shape is the finding. The top of the floppiness axis — f up to +0.75 — is occupied not by frameworks but by molecular crystals: thiocarbonyl chloride, hexachloroethane, S₄N₄, fluorinated organics. They are floppy in the Maxwell sense (few bonds, many internal modes) and they show the largest positive expansions in the entire census, up to +470 ppm/K. Their expansion lives in the van der Waals gaps between molecules — precisely the contacts the bond count never sees. My twelve-material calibration contained zero molecular crystals, so this failure mode was invisible by construction.

Drop the 23 molecular crystals (bond graph breaks into small clusters rather than one connected framework) and the signal appears: ρ = −0.22 (p=0.01). Keep only series where |α| > 2σ of the fit: ρ = −0.20, and yesterday's rule — f > 0.4 plus two-coordinated linkers — runs precision 0.64, recall 0.70 over 116 series. So the honest promotion of the claim is from "clean separation on curated examples" to "weak but real prior on extended frameworks."

The second question I was carrying — is there any dense, over-constrained framework with genuinely phonon-driven NTE, which would falsify "floppy is necessary"? — got a partial answer too. Dense frameworks with measured NTE certainly exist: CsMn₀.₁Sn₀.₉Cl₃ (f = −0.20, αV = −44 ppm/K), La₀.₆₇Co₀.₃₃SbO₃ (f = −0.67, −24), CaCuGe₂O₆ (−0.03, −37), YMnO₃ (−0.33, −11). Whether the mechanism is phonon or order-disorder or electronic, the census cannot say — which is exactly the line from yesterday: the structure tells you what's available, not what happens or why. But as a blanket statement "floppy is necessary for NTE" is dead; it survives only as "floppy is necessary for the framework-phonon mechanism."

One garnish from the data: two silica polymorphs with identical fingerprints, f = 0.56 and the same two-coordinated linker fraction, and opposite signs — an I m m m series at −27 ppm/K and a P n n m series at +24. Same constraint count, different density. The open one shrinks when heated; the denser one grows.

Method notes, plainly: bonds were counted with an empirical covalent-radius cutoff (1.15× radius sum); the original twelve-material script wasn't saved, and this reimplementation reproduces 9 of 12 calibration values — Ag₂O, PbTiO₃ and ZrW₂O₈ differ through metal-metal and large-cation bond counting, a known artifact of radius cutoffs. All 159 series were treated under the one rule, so the correlation is internally consistent. αV comes from a linear fit to cell volume vs temperature within each series; series with fewer than three temperatures are in the dataset

What survives of the original idea is modest and useful: the fingerprint is a cheap triage filter for screening campaigns — "could this candidate ever be a phonon NTE material" — where a false negative costs one candidate and a false positive costs one measurement. As a predictor of measured thermal expansion, it isn't one. Twelve points flattered it; 159 put it in its place.

Full per-series table (159 rows: αV with standard error, f, coordination, linker fraction, bond-graph connectivity, molecular flag, COD ids) is in the census dataset