An identification of three constructions of one lattice, with the closed transverse trace as its lock (Geometry of the vacuum, floor −1)

Abstract

A cell of d+1 unit directions with pairwise Gram -1/d arises by a representation-theoretic route (irreducibility of the standard representation of the symmetric group on d+1 letters, then Schur), with no energy, no Hamiltonian and no minimization anywhere in the chain; the single residual freedom is a unit of length. We show by computation that this cell is the same object as two constructions already standing in the literature: the canonical placement (standard realization) of a topological crystal, whose characterizing conditions (zero sum at a vertex, tight frame) are satisfied by exactly this Gram, and the lattice on which the nodal-semimetal literature computes. The two corpora do not cite each other. On that lattice the two-component symbol linearizes at a generic zero into a node-independent Pauli form for all d (the high-symmetry points, where the rank drops, are a separate named locus), and the transverse metric has rank 2 with d-2 flat directions parametrizing the zero set. The trace of that metric is carried by the Gram alone, hence is constant along the entire zero set and equals (d+1)^2/d, so the transverse trace is (d+1)^2/(2d); this reproduces, exactly for every d = 2..6 including the d = 2 anchor, the closed form published for the same family, which the spectral corpus obtains from a Hamiltonian. The number is not claimed as new (the closed form is published, 2024) and the two derivations share their key step, which is stated openly; it is used as the lock of the identification. The lattice of the standard representation, the variational derivation of the standard realization, the codimension-2 zero set for all d and the closed velocity are each cited, not claimed. Statements about rank and the Pauli form are for generic zeros; the trace identity is symbolic in d, and the run over d = 2..6 is a control, not the proof. No physical reading of any object is made or implied.