Crystallographic Closure of Integral Lorentzian Lattices and the Rank at Which It Ends (Geometry of the vacuum, floor 0)
Year: 2026
Type: preprint
Abstract: For an integral lattice of signature (n,1), every finite-order isometry is shown to fix an INTEGRAL timelike vector up to sign, by averaging one integral timelike vector over the finite orbit of the isometry after the sign that preserves a cone component has been factored out. The orthogonal complement of that vector is then a positive definite integral lattice of rank n, invariant under the isometry, and the set of finite orders realisable on signature (n,1) is proved EQUAL -- both inclusions -- to the set realisable on positive definite lattices of rank n. For n = 3 the trace argument on the definite complement gives orders {1,2,3,4,6} for every integral lattice of signature (3,1), unimodularity playing no role in the bound; for the specific lattice I_(3,1) all five orders are exhibited by explicit integral matrices, so the set is sharp. The mechanism is separated into its three independent ingredients -- negative index one, integrality, and rank four -- and each is shown to be load-bearing: signature alone does not forbid order five, since a real rotation by 2*pi/5 preserves the form and fails only to be integral. A parity lemma, derived from the structure of invariant forms on isotypic components, states that a rotation component always contributes an even negative index, so an odd negative index forces a plus-or-minus one eigenvector. The question of which non-crystallographic orders act on a UNIMODULAR lattice of rank four is settled for every lattice at once by the square class of the determinant of the family of invariant forms, an invariant independent of the choice of lattice inside a fixed rational representation: orders five and ten are excluded from every unimodular lattice, order eight acts on I_(2,2) with an explicit order-eight matrix given, and order twelve admits an even unimodular form of signature (2,2) while the odd case is left open. Finally the orders realisable on signature (n-1,1) are tabulated for ranks two to eight, showing that the crystallographic set survives up to signature (3,1) and breaks at (4,1), where orders 5, 8, 10 and 12 enter, with the next break at (6,1). No physical reading of any object is made anywhere in the work.
Author: Sobol, Vladimir
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204472367
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