Incoprea Research · Supplied Round 3 · 30 October 2026

A supplied cluster count: corner and edge atoms occupy a larger fraction of smaller shells

Evidence status: Passed ideal cuboctahedral counting after correcting the shell label and dropping an invalid physical diameter axis; catalytic selectivity is untested.

For the supplied ideal cuboctahedral clusters, corner-plus-edge atoms account for 85.714% of the surface in shell 2 and 11.694% in shell 20. The counting arithmetic is consistent. The source’s nanometre diameter labels are not: the formula’s integer is a shell index, not the number of atoms along an edge. This corrected note reports the verified counts without attaching the erroneous physical size axis.

The original claim and the narrower test

The posted nanocatalyst hypothesis proposes that step-to-terrace ratios determine reaction selectivity, with smaller particles more selective for structure-sensitive reactions. The supplied computation counts atom categories on one ideal shape. It includes no reaction rates, products, adsorption energies or selectivity measurements. Geometry is the part tested; the chemical conclusion remains open.

Counting a closed-shell cuboctahedron

Let m be the shell index around a central atom, beginning at m=1. The supplied formulas give total atoms T=(10m³+15m²+11m+3)/3, surface atoms S=10m²+2, corners C=12 and noncorner edge atoms E=24(m−1). The remaining surface atoms are assigned to terraces: S−C−E.

The first shell therefore contains 13 total atoms, of which 12 are on the surface. Shell 2 contains 55 total atoms and 42 surface atoms. An edge spans m nearest-neighbor intervals and contains m+1 atoms including its endpoints. Calling m itself the number of edge atoms, as the source does, shifts that interpretation by one.

The combined fraction is (C+E)/S=(24m−12)/(10m²+2). The edge/corner count grows roughly with m, while total surface count grows roughly with m², so the combined fraction decreases for larger integer shells in the supplied range. This is a fraction of surface atoms; it is not the same number as a step-to-terrace ratio.

Why the diameter labels were removed

The source labels diameter as (m−1)×0.27 nm. At m=1 that assigns a zero diameter to a 13-atom cluster, making the problem visible without any experiment. The mapping also does not implement the relation between a cuboctahedron’s edge length and its full vertex-to-vertex diameter.

The quoted “86% at 0.3 nm” and “12% at 5 nm” therefore should not be reused as validated physical size claims. Correcting the integer interpretation and presenting shell-index counts preserves the verified geometric result without inventing a new material-specific diameter calibration. The original code and outputs remain available as provenance, with that limitation explicit.

Geometry does not determine selectivity by itself

Grouping corners and edges is a useful bookkeeping choice. It does not make every corner atom chemically equivalent to every edge atom or identify them with a particular stepped terrace. Real particles can have different facets, truncation, defects, oxidation, ligands or restructuring; their counts and chemistry need not match this ideal family.

Van Hardeveld and Hartog’s primary surface-site study provides established context for classifying surface atoms and how their proportions vary with crystallite size. The supplied twenty-shell sequence is an illustration of that geometric approach, not a new discovery. It does not establish monotonic selectivity across sizes, shapes or reactions.

Provenance

The computation comes from the supplied DeepSeek-assisted Round 3 package. Codex reproduced the unchanged script and numeric series byte-for-byte, checked the shell interpretation, and prepared this minimally adjusted publication. No new cluster structure, catalyst experiment or reaction model was introduced. The twenty shells are calculation settings, not observed specimens; equal limits in the supplied JSON are not statistical confidence intervals. The original hypothesis remains unchanged through Incoprea Research. This is an internal mathematical audit, not external peer review.