Perspectives

Property

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to turn

–held-out accuracy

0 of reactions

Press Train: the map learns from the reactions one by one.

held-out accuracy (learned so far)rangebits saved

Rotate the frame, compress the data

One arrangement of 94 points, read from 18 directions. Every element gets a single position. Each property is not stored as a column of numbers but as a direction through that same cloud: turn the sphere until a property's plane is the equator, and latitude reads as that property. Choosing a property is simply rotating the reference frame.

~90%of element pairs put in the right order for a property the map never saw (held out; pairs weighted by measurement confidence)

99.9%fewer bits than the 41,285 reaction records it learned from

Why rotation compresses. The 18 properties are not independent: electronegativity, ionization energy, the radii and the thermal properties lean in related directions. A fixed set of axes would need a column per property. A shared arrangement plus a rotatable frame needs only the positions (3 numbers per element) and one direction per property: 221 bytes at 5 bits per number, against 0.48 MB of interaction records. It is also more accurate than reading the raw interactions directly: an element's co-occurrence counts give 71% held-out accuracy, the map's positions alone 83%.

How the arrangement is found. Elements start at random places; no periodic table is imposed. Reactions alone move each element towards the elements it actually combines with, and the arrangement grows organically. Every retraining uses a different random start and a different order of the 41,285 reactions (three orders are prepared), so the final arrangement differs a little each time while the chemistry it captures stays the same: the map's positions alone reach 83.6–85.8% held-out accuracy, and with two crystal-structure vectors, five element-data sources, extra measured properties and isotope descriptors 89.6–89.9%. Starting from the periodic table instead gives 85.8% and 89.9%: the table is a helpful head start, not a requirement. Values are read half from the property's plane and half from the element's nearest neighbours.

Where this is useful: see Applications below.

Limits. Not a source of design values: where a wrong call is costly rather than cheap, for exact properties, safety-critical decisions or conditions outside the data (high pressure, unusual oxidation states, compounds rather than elements), measure or compute directly. Accuracy here is pair order (50% = chance), not exact values: held-out R² is 0.63–0.68. Atomic mass, electron affinity, heat capacity and thermal conductivity sit poorly in any shared frame (60–84%). The 99.9% compares against the records as stored, not their best possible encoding.

Applications

The frame is strongest at ranking and similarity: it orders element pairs correctly about 90% of the time, puts interchangeable elements side by side, and estimates a compound's stability to about ±0.35 eV/atom. It decides what to try first; experiments or exact calculations make the final call. It pays off most at the edge, where decisions must be instant and resources minimal, and an 85%-right call now beats a perfect one later.

From the element frame

Edge decisions: fast, tiny, offline. The whole frame is about 0.35 KB, small enough for any microcontroller, sensor or handheld instrument, with no network or database. Reading a property is a single dot product (3–6 multiply-adds per element; a compound is the average of its elements), so answers take microseconds and almost no energy. Typical uses: sorting scrap and recycled metals or batteries on a conveyor from a portable X-ray fluorescence or laser-spark reading; triaging ore and alloy assays in the field; choosing the next experiment in a self-driving lab; adjusting a dopant or furnace recipe in-line. In each case a wrong call is cheap and is caught later, while waiting for an exact value is not an option.
Element substitution. Find elements that sit close to a costly, scarce or toxic one in the same direction that matters, e.g. alternatives to cobalt in battery cathodes or to lead in perovskites, and rank them before any synthesis.
Multi-property screening. Because each property is a direction, trade-offs are angles between planes: an element that sits high on one plane and low on another (light but stiff, conductive but stable) can be found geometrically, and Pareto-style shortlists drawn in seconds.
Synthesis planning. The same positions score which elements and precursors tend to combine, prioritising partner and precursor choices for a target compound and cutting the number of trial reactions.
Data checking. A measured value far from its modelled place (a long error offset) flags a likely error, unit slip or a different allotrope or state, as with carbon's bulk modulus here; the same test flags implausible text-mined reactions.
First estimates for gaps. Rough, ranked estimates for elements or properties that are poorly measured (heavy, rare or radioactive elements), to decide which measurements are worth making.
Compact inputs for other models. A compound's averaged element positions give 3–6 numbers per element that carry periodic and reaction chemistry, useful features for small-data machine learning.
Teaching. Periodic trends become rotations of one object: turn to a property and its trend is latitude, which shows why electronegativity, ionization energy and radius move together and why electron affinity does not.
Beyond chemistry. The method applies wherever entities interact and carry measured attributes: ingredients and recipes, genes and their interactions, products and purchases, materials and processes. One shared arrangement learned from the interactions, with each attribute as a direction, compresses and connects the data the same way.

From compound vectors

Stability triage in microseconds. A compound's predicted formation energy needs only its formula: average the element positions, add the squared-distance interaction, about a hundred multiply-adds. Millions of hypothetical compositions can be ranked on a laptop, or on an instrument, before any expensive quantum calculation, which is then spent only on the promising few.
Reaction driving force. The same vectors give every product and precursor an energy, so a route's driving force (product minus precursors) can be estimated instantly and synthesis routes ranked by how strongly they are pulled forward.
Why a compound is stable. The energy splits by axis: how much comes from charge transfer (the halogen/oxygen ⟷ alkali axis) and how much from size or covalent contrast. That points to which element to swap, and in which direction, to make a compound more or less stable.
Dopants and solid solutions. An element close to the one it replaces adds little interaction energy, so it is likely to dissolve; one far away along a binding axis is likely to form a new phase instead. A quick first screen for doping and alloying.
Checking computed and measured databases. A compound whose recorded energy sits far from its vector prediction is worth a second look: an unusual structure, a calculation problem, or genuinely surprising chemistry.
Compact compound fingerprints. A compound's vector (where its elements sit) and spread (how far apart they are) give a handful of numbers that carry periodic, reaction and bonding information, ready as inputs to other models.
Beyond chemistry. Interaction strength as squared distance along learned axes applies wherever things combine: flavour pairings, drug combinations, protein partners, team or supply-chain matching. Place the parts in one shared space, and how well a combination holds together is read from how far apart its parts are, along which directions.

Not for exact design values, safety-critical decisions, or conditions outside the data (high pressure, unusual oxidation states). Compound energies are weakest for intermetallics and underestimate strongly ionic oxides. The driving-force and dopant uses follow from the same algebra but have not yet been tested against data here.

Compounds as vectors

A compound gets its own vector: the average of its elements' positions, weighted by how many atoms of each it contains. How far apart those elements sit in the map then gives its interaction energy: formation energy = each element's own contribution, plus one learned term from the squared distances between the compound's elements along each axis. It is Pauling's rule (bond energy grows with the squared electronegativity difference), extended to every direction of the map.

Formation energy of unseen compounds	Error	R²

95,559 Materials Project compositions (most stable form of each, 2019 snapshot); 10% held out; error in eV per atom. The map never saw a formation energy: its positions come from reactions and element properties.

Formula

Band gaps. Against 4,602 measured band gaps (Zhuo et al., 2018), a compound's three map coordinates alone order the gaps of non-metals correctly 75% of the time (rank agreement 0.69), slightly better than 36 averaged measured element properties (73%). Telling metals from non-metals needs more: position alone gets 69% right, position plus the spread of the compound's elements 80%, measured element data 85% (always guessing the commoner class: 53%). Position carries how strongly electrons are held on average; whether a gap opens at all depends on how different the partners are.

What the plane angles reveal

Two properties whose planes are parallel rise and fall together; at 90° they are independent; at 180° they oppose. Each link turns the sphere so the two planes are seen edge-on, crossing at their angle.

Three directions carry the 18 properties. 75% of the plane directions lie along one axis (electronegativity, ionization energy, electron affinity ⟷ size, polarizability), 16% along a second (size ⟷ stiffness and cohesion) and 9% along a third (bond strength and heat transport ⟷ heaviness). electronegativity vs covalent radius
Known laws reappear. Heat of evaporation tracks boiling point (Trouton's rule, ~10°), melting tracks cohesive energy (~10°), electronegativity tracks ionization energy (Mulliken, ~19°). Trouton · Mulliken
A link the raw table hides. Heat capacity looks unrelated to melting point (r −0.06), yet the planes tie them. Hold size fixed and the link appears (−0.62): stiff, light, high-melting solids (C, B, Be, Si) sit far below the classical 3R at room temperature, the Debye effect. heat capacity vs melting point
Reactions aligned heat transport with electron donation. At the periodic-table start, the thermal-conductivity plane sat at about 120° to electronegativity; after learning the reactions it sits at about 160°, almost opposite. The elements that give away electrons in reactions are the ones whose free electrons carry heat (Wiedemann–Franz). thermal conductivity vs electronegativity
Electron affinity stands apart. Its plane is close to 90° from atomic mass, density and bulk modulus: it is set by shell filling, not by bulk or mechanical behaviour, which is why it is among the hardest properties to predict. electron affinity vs bulk modulus

The planes were fitted to these same properties, so most angles mirror known correlations (r = 0.94 between angles and correlations). "Hidden" means not visible in the raw table, not unknown to physics: both findings match established theory.

What the residuals show

A residual is the gap between where the map places an element and its measured value. On the globe each Residuals line joins the element's place in the map (Model view) to its measured value (Actual view); switching views slides the element along its own line while the line stays still. Long lines mark elements the map places poorly for the chosen property, and they shorten as the map learns. The accuracy figures go one step further and use held-out predictions, read for each element while its value was hidden. Most are small. The largest ones, where two independent read-outs agree, fall into three kinds.

Weak or estimated data. Where a second, independent source disagrees with the table, that source sides with the map in 16 of 20 large misses, against 28% for ordinary misses. The flagged values are mostly extrapolations or old compilations: francium and radium radii (barely measurable, highly radioactive), Bondi's 1964 van der Waals radii for caesium and lithium (later revisions move toward the map), neon's Pauling electronegativity (neon forms no compounds) and xenon's electron affinity (unbound anion; now removed). Checking thermal values also turned up a wrong neon melting point (48 K instead of 24.56 K), now corrected.
Known physics a three-axis map cannot hold. Hydrogen (no inner shell, fits no group) has the most misses. Then the closed shells of helium and neon, the first-row anomaly of lithium, carbon, nitrogen and fluorine (no inner p-shell, so unusually small and electronegative), the low room-temperature heat capacity of stiff light solids (carbon, boron: the Debye effect), gadolinium's magnetic transition at 293 K, and gallium's paired-atom solid.
Model limits. Atomic mass is known almost exactly, yet light elements missed by several error widths: mass is nuclear, not chemical. Adding descriptors from all 3,186 known isotopes cut these misses (atomic mass 87.8% → 91.1%).

Compounds. Against 95,559 computed formation energies (Materials Project, 2019 snapshot), the map's compound model flags 1,837 entries above +1 eV/atom, many of them physically implausible: Ta₂TiMo at +4.6 eV/atom, where refractory metals mix at close to zero. The map puts this group at −0.17 eV/atom (median). Three independent databases confirm the flag: for the same compositions OQMD gives a median of −0.24 eV/atom (1,587 found, 88% below +0.5), Alexandria −0.44 (216 found, all below +0.5) and JARVIS −0.07 (799 found), while on ordinary compositions the databases agree to within 0.01–0.07 eV/atom. Refitting without these entries improves the fit for everything else (held-out error 0.312 → 0.289 eV/atom).

What the map cannot do. Against 2,699 calorimetric measurements, the calculations are far closer to experiment (mean error 0.11–0.14 eV/atom for OQMD, Alexandria, JARVIS and Materials Project) than the map's compound model (0.31). Where the map and Materials Project disagree by more than 0.3 eV/atom, the calculation is the closer one 90% of the time. The smaller family offsets the map shows (transition-metal oxides, lanthanides, actinides, phosphides and borides) are therefore its own limits, not calculation errors: experiment puts Materials Project within 0.01–0.07 eV/atom for those families. The map is good at catching gross errors and implausible entries, not at refining good values.

None of the large residuals points to new chemistry: each has a known explanation. Their value is as a consistency check that flags weak or implausible data, including gross errors in computed databases (confirmed above by three other databases), without being told which entries are estimates.

Property

Held-out accuracy of this property plane—

0%50% coin flip100% goal

One arrangement for every property?

Each property alone would prefer its own layout. Giving every property a plane through one shared map, and letting training tilt the planes, finds one arrangement that serves them all. The planes merge into a few shared directions. The radii and polarizability point the same way, and so do melting and boiling point. Ionization energy and electronegativity point the opposite way to radius and mass.

Three dimensions can't hold everything. Across the 18 properties, 3 directions capture of the variation, 6 capture and 8 capture .

Click a property to turn the sphere to it. The weights set how much each property counts toward the score next to the title (0 leaves it out). By default, the properties that govern bonding and reactivity count most: electronegativity (Pauling) and ionization energy 3; electron affinity, atomic and covalent radius, melting point and heat of formation 2; everything else 1. Your changes are remembered in this browser. Now: .

Pushing toward 100%

Held-out share of element pairs in the right order (%). Earlier rows count every pair equally; the last row weights each pair by measurement confidence, as the headline does.

Approach	EN	IE	Melt	Mass	All 18	Rxn
3D, reactions only	79	75	71	64	–	69
12D, reactions only, 4 nearest neighbours	84	76	78	72	–	75
200D SkipAtom (crystal structures)	89	82	81	73	–	–
Previous map: 3D, reactions + 18 properties, planes only	88	85	86	70	80	67
Structure vectors only (CHGNet + CrystaLLM), no map	–				81	–
Structure vectors + five element-data sources, no map	–				85	–
This map, every pair counted equally	91	91	88	91	87	67
This map

Each value is read half from a plane and half from the element's 5 nearest neighbours that have a measured value, using the map position plus two structure-learned element vectors (CHGNet, CrystaLLM; 8 principal components each) and five element-data sources: electron configuration, oxidation states, chemical ordering scales (Mendeleev, Pettifor, Glawe numbers), geochemical class and lattice type, and calculated atomic-orbital energies. None contains a scored property; the outermost orbital energy is a close relative of first ionization energy. Atomic mass is on a power scale close to a square root; electron affinity is scored only for the 72 elements with a measured value, and heat capacity per mole of atoms. A 6-dimensional map was tried and dropped: it gave no held-out gain (86.2% vs 86.3%) at twice the size. Adding 15 more property planes, or just the five that point into the least-covered directions, also gave no gain (86.4% vs 86.9%).

EN electronegativity (Pauling) · IE first ionization energy · Melt melting point · All 18: the weighted average over every property, the figure shown next to the title (it includes hard properties such as electron affinity, heat capacity and thermal conductivity, so it sits below the four shown here) · Rxn reaction test, not part of the average. In the last row each hidden value is predicted from the element's reactions and its other properties; near-duplicates are hidden with it.

How position decides who reacts with whom

Reactions are scored by one shared rule, fit(a, b) = position a · M · position b, measured from the sphere's centre. Along an axis of M with a negative weight, opposites pair up; along a positive one, like pairs with like.

Hover any element to draw lines to its six likeliest partners.

Domains

Each dot sits at the plain average of its elements' positions, and 93% of the compounds contain oxygen, which sits near one edge of the element cloud. Averaging pulls every oxide toward oxygen by a fixed share: a binary oxide sits halfway to it, a three-element compound a third of the way, a four-element one a quarter. So the dots stack into parallel layers, one per number of elements, and seen edge-on they look like interference fringes. Element count and oxygen explain about three-quarters of a dot's layer position on the earlier 3D map, and about half on the current one, whose layers are softer.

Within a layer, the elements of each family lie on near-parallel lines, so swapping an element for a family-mate slides a compound along a streak. The densest streaks are the dataset's research hot-spots: Sr/Ba/Ca/La compounds (perovskites), Li–Mn–P compounds (battery cathodes) and P/Fe/Al/Bi compounds (phosphates, ferrites). The pattern comes from the averaging, not from a new physical law.

Filled and empty domains. Not every part of the sphere can be occupied. A compound is an average of its elements, so it must lie inside the hull the 94 elements span (only 6% of the sphere). Inside that hull, comparing 95,559 real compositions with the same compositions built from randomly swapped elements shows where chemistry crowds in and where it stays away: closed shells (noble gases), donors without acceptors (alkali-rich mixes) and late transition metals with carbon or boron stay empty; ionic halides, oxides and refractory carbides are over-filled.

◎ Show filled and empty domains marks the volumes on the globe (green filled, red empty, violet closed shells); move the mouse over one to read why it exists.

↻ Align the view with the bands turns and zooms the sphere so the layers are seen edge-on, oxygen side at the top, and names each layer with its commonest compositions.

Process arrows

Each arrow points toward the compounds that more often needed that process: heating to a higher top temperature, cooling or quenching, and pressing powder into pellets. An earlier test found these are weak signals, because how a compound is made depends mostly on its precursors and the lab.

Data sources

Reactions. 31,782 solid-state and 9,518 sol-gel syntheses text-mined from published papers (Kononova et al., Scientific Data 2019; Ceder group, github.com/CederGroupHub/text-mined-synthesis_public), streamed once in a fixed shuffled order. reactions have a product with two or more elements up to plutonium, making distinct element combinations. Pressing means compacting powders into pellets ( reactions); cooling means an explicit cooling or quenching step (); top heating temperatures were reported for reactions.

The 18 scored element properties. XenonPy element table (Yamada et al., 2019), read through the ElementEmbeddings package (Onwuli et al., 2023). Each property is placed on its own power scale (Box–Cox), chosen without looking at the predictions so that its values spread symmetrically: close to a log scale (like pH) for the electronegativities, electron affinity, heat of formation and thermal conductivity; stronger than log for first ionization energy, whose few huge values (He, Ne, F) would otherwise dominate; about a square root for the radii, melting and boiling points, density and the heats of fusion and evaporation; and stretched (power above 1) for heat capacity and covalent radius, whose values bunch at the top. A single log scale for everything scores 1.3 points worse; these per-property scales score 0.6 points better and lift the map's own positions from 84.2% to 85.8%. Confidence weights are computed in each property's original units, so the comparison is fair. Two definitions are cleaned: electron affinity is kept only for the 71 elements with a measured value in the mendeleev database (23 filled-in or unbound-anion estimates removed, e.g. He −19.7 eV, Xe −0.056 eV, U 3.1 eV), and heat capacity is per mole of atoms (H₂, N₂, O₂, F₂, Cl₂, Br₂ and I₂ halved).

Heating and cooling properties at 1 atm. Melting point and heat of fusion are kept only where the element melts at atmospheric pressure: helium (solidifies only above 2.5 MPa), carbon and arsenic (both sublime; their tabulated melting values were measured under pressure) are left out. Boiling point is the temperature at which the element turns to gas at 1 atm, so carbon's is its sublimation point, 3915 K (the tabulated 5100 K is a high-pressure value). Heat of evaporation is per mole of atoms, like heat capacity (the seven diatomic elements were per mole of molecules, and the nitrogen and oxygen entries were wrong). Two table errors found on the way were corrected: neon melts at 24.56 K, not 48 K, and its heat of fusion is 0.335 kJ/mol, not 0.164.

Read-out sources (none contains a scored property):

Crystal-structure vectors: CHGNet (Deng et al., 2023) and CrystaLLM (Antunes et al., 2024) element embeddings, via ElementEmbeddings; 8 principal components each.
Electron configuration, oxidation states, Mendeleev / Pettifor / Glawe numbers, geochemical class, lattice type: mendeleev Python package (Mentel, 2014–2025).
Isotopes: all 3,186 known isotopes of these 94 elements in the mendeleev isotope table (masses, abundances, half-lives, spins), summarised per element as the number of known and stable isotopes, the half-life of the longest-lived isotope, the spread of isotope masses in natural abundance, and the nuclear spin of the main isotope. They are also drawn on the map: each element carries a ring of its isotopes in order of mass number, stable ones larger and more opaque, radioactive ones fainter the shorter they live (hover one for its half-life and abundance). Chemistry does not tell isotopes apart, so they share their element's place. No isotope mass is used directly, so atomic mass is still predicted, not looked up. They add 0.1 points overall and lift atomic mass from 87.8% to 91.1%.
More measured element properties: work function, Debye temperature, thermal expansion, Miedema's electronegativity and electron density, and magnetic moment, from the elementy package (Forrest; compiled from standard handbooks). They raise held-out accuracy by 0.5 points, most for heat of fusion (+2.6), thermal conductivity (+1.9) and melting point (+1.7), the properties tied to how stiffly atoms are bound.
Test only: Rahm–Zeng–Hoffmann electronegativity (average valence-electron binding energy, elements 1–96; J. Am. Chem. Soc. 2019). The map never saw it, yet predicts it held out with rank agreement 0.85 from positions alone (0.89 with element data), against 0.89–0.94 for the two electronegativity scales it was trained on. The largest disagreements (Gd, the noble gases, H) are where that scale's own authors note its dependence on the chosen valence state.
Second compilation: PubChem periodic table (U.S. National Library of Medicine, Nov 2026), compared value by value with the training table. 51–98% of entries are identical copies; where they differ, PubChem matched standard references (CRC, NIST) for 32 values that were wrong in the training table and these were corrected: melting points of Li (553.7 K → 453.65 K), B, Cr, Pm, Ir and Pa; boiling points of 19 elements, among them Li (1118 → 1615 K), Si (2628 → 3538 K), Be, Fe, Ni, Sn, Ba, Hf and W; astatine's ionization energy (11.15 → 9.32 eV, measured 2013); gallium's electron affinity; and the Pauling electronegativity of W, Lu and U. Where PubChem disagrees on heavy-metal ionization energies (Ta, Os, Th, Fr) it is the one rounding, and the table was kept.
Test only: SuperCon2 (Foppiano et al., NIMS, 2022; 40,324 superconductor records text-mined from 37,700 papers). Of 2,237 compositions with a critical temperature measured at ambient pressure, a compound's three map coordinates order the temperatures correctly 68% of the time (rank agreement 0.48), measured element data 72%. Superconductivity depends on electronic and lattice details beyond an average position; high-temperature superconductors (36% iron-based, 22% cuprates among those above 30 K) sit apart from the rest by about half the spread of all compounds.
Cross-checks for compounds (not used in training): OQMD (Kirklin et al., 2015; 394,993 compositions), Alexandria (Schmidt et al.; 116,623 stable compositions, formation energies derived from its own elemental references) and JARVIS-DFT (Choudhary et al., 2025 release; 65,233), all via the JARVIS-Tools collection; experimental formation enthalpies from high-temperature calorimetry (Kim, Meschel, Nash and Chen, Scientific Data 2017; 1,131) and the SGTE SSUB database (1,643); experimental band gaps (Zhuo, Mansouri Tehrani and Brgoch, 2018; 4,602 compositions).
Calculated atomic-orbital energies: pymatgen (Ong et al., 2013), from NIST atomic reference data (density-functional calculations).

Extra training planes (tested as additional perspectives, never scored): orbital energies, oxidation states and ordering scales as above; sound velocity, thermal expansion and mineral hardness from pymatgen's element table; calculated band gap and magnetic moment of each element's solid from the Magpie element data (Ward et al., 2016, OQMD calculations) shipped with matminer.

Compound energies. Materials Project formation energies, 2019.04.01 snapshot (133,420 structures; Chen et al., MEGNet, figshare 8097992), reduced to 95,559 compositions using each one's most stable form.

Benchmarks only (shown in tables, not used by the map): SkipAtom (Antunes et al., 2022), Mat2Vec (Tshitoyan et al., 2019) and ORB element vectors, via ElementEmbeddings.

Method

Each element has a free 3D position inside a sphere of radius 6, starting at its cell on the periodic-table spiral; what you see is the map itself, with no projection. The view starts on the periodic table and turns to the selected property's plane. Reactions train by substitution: each element of a compound is predicted from the others. Property planes are fitted at the same time, and their errors pull on the positions. The reactions' pull fades as training goes on. Held-out accuracy comes from 5 runs, each hiding a different fifth of the element–property values; the map shown is a sixth run that sees everything. Positions between saved checkpoints are interpolated.

R². Next to each pair-order figure the page shows held-out R²: one minus the sum of squared residuals over the total variation, from the same held-out predictions, on each property's display scale and averaged with the same importance weights. Unlike pair order it counts how far each prediction is off, so a few large misses weigh more; it is lower and starts near zero for random positions.

Range. Every accuracy carries an upside and downside range. Two things move it: errors in the measured values (each value redrawn within its typical uncertainty, the same one used for confidence weighting) and which elements happen to be measured (the 94 resampled with replacement). The range is ±1.96 standard deviations of the combined change, widened on the upside by the score lost to measurement noise: noisier data drags a good model's agreement down by about 0.7 points on average, so cleaner data would most likely raise it. At the end of training: 89.6–89.9% depending on the reaction order, each with a range of about 88.5–91.7%. Random measurement error alone moves it by about ±0.2 points; which elements are measured by about ±1.1; the reaction order by about ±0.15. Gross errors in the source table mattered more than random noise: correcting 32 values that a second compilation (PubChem) showed to be wrong raised the score by 0.25 points, so a few bad values do not hide a much better model.

How accuracy is weighted. Each property carries an importance weight (3 for Pauling electronegativity and first ionization energy, 2 for electron affinity, two radii, melting point and heat of formation, 1 otherwise). Within a property, each pair of elements counts by how surely its measured order is the true order: 2Φ(|Δ|/√2σ) − 1, where Δ is the measured difference and σ the typical experimental or definitional uncertainty of that property (for example 0.01 eV for ionization energy, 15 pm for van der Waals radius, 5–12% for heats of fusion and thermal conductivity). Pairs inside the noise, which no model could order reliably, count little. This lifts the score by about 1.3 points over counting all pairs equally, mostly through the radii and Pauling electronegativity; comparisons quoted elsewhere on this page (6D, extra planes, element data alone) use equal pair weights.