bibkey: conwaysloane1999spherepackings
authors: J. H. Conway and N. J. A. Sloane
year: 1999
title: Sphere Packings, Lattices and Groups
doi: 10.1007/978-1-4757-6568-7
claim: Classical root-lattice and dual-lattice geometry, including the A-type simplex lattice; the local Lean proof establishes its integer minimum-shell formulation.
strata_touched: []
license: citation-only
triage: anchor

A-type dual lattice and simplex energy

The classical dual of the root lattice A_r is the projection of the integer coordinate lattice onto the hyperplane with coordinate sum zero. For integer y with r plus one coordinates, the squared norm of that projection is sum(y_i squared) minus sum(y_i) squared divided by r plus one. Multiplying by r plus one gives the unordered pair-difference energy. Translation by a constant integer vector preserves the projected vector and this energy.

The local SimplexMinimalShell.complete_shell proves the all-integer minimum and equality reconstruction directly: the minimum nonzero energy is r, and equality consists of a signed singleton perturbation of a constant vector. Setting one coordinate to zero gives the signed coordinate units and signed all-ones charges. This is classical A-type dual minimum-vector geometry, with no novelty claim.

Verified locator

The book identity, authors, year and DOI 10.1007/978-1-4757-6568-7 are confirmed by the Crossref record: https://api.crossref.org/works/10.1007%2F978-1-4757-6568-7 . The root/dual terminology and book reference are independently displayed at https://errorcorrectionzoo.org/c/an and https://errorcorrectionzoo.org/c/an_dual . Those web entries establish the lattice identification, not a locator for the complete equality proof. The book’s full text was not inspected; the integer cut-energy proof and its exact quantifiers are supplied by the local Lean source.