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hexagon:2611.00089v1

math.LO — Logic

Inertia and Elementary Theories of Hermitian Nilpotent Operators

Contributed by Zhenyu Sun, Jilin Wang

Let VV be a finite-dimensional Hermitian space over an algebraically closed field with a nonidentity involution, and suppose its form hh is nondegenerate. For a self-adjoint nilpotent operator NN, we recover the multiplicity of each signed Jordan block from finite differences of the ranks and signature differences of the forms h(x,Njy)h(x,N^j y). On a fixed Hermitian space these inertias classify unitary conjugacy. Across scalar fields they classify elementary equivalence in the four-sorted language of scalars, vectors, endomorphisms, and projective points with NN named. At fixed dimension each admissible inertia sequence isolates a complete theory. These theories are model complete and decidable, and none has the independence property. A bivariate generating function counts them by ambient signature. For two self-adjoint square-zero operators with a common image of fixed positive dimension and positive definite quotient forms, simultaneous unitary conjugacy on a fixed Hermitian space is classified by the ordered generalized eigenvalues. The associated Hermitian pencil recovers these eigenvalues even when all operator words have constant ranks and inertias. At fixed signature and rank the family theory is model complete and decidable. Its completions correspond to the ordered-field types of the spectrum, and each completion has the Turing degree of its type. We also characterize the Witt index by square-zero ranks over arbitrary fields with involution. For positive definite spaces we prove that finite-measurement sentences at fixed dimensions and outcome count have the same truth value over R(−1)R(\sqrt{-1}) as over C\mathbb C for every real closed field RR. The least Naimark dilation dimension of a measurement is the sum of its effect ranks.

Primarily human-written textHuman understanding: some partsHermitian forminertiamodel completenessquantum measurementreal closed fieldself-adjoint nilpotent operator
Nov 04, 2026PDF ↗View work →
hexagon:2611.00078v1

math.NT — Number Theory

GRAM OBSERVATIONS AND THETA-SERIES RECONSTRUCTION OF EUCLIDEAN LATTICES

Contributed by Changxia Lin

A finite collection of lattice-vector lengths probes a positive-definite Gram matrix through rank-one linear functionals. We use this observation map to give a local criterion for non-identifiability and to construct an exact three-dimensional collision at a closed squared-length cutoff of one: two non-isometric lattices have identical complete truncated vector-length data, although their Gram determinants are different. A signed-cycle construction gives a higherdimensional family whose complete rank-one length multisets agree at every cutoff below an explicit threshold, with an exact first-shell discrepancy. We then turn to multi-vector data. The full degree-n Gram theta data determines every rank-n Euclidean lattice, by a basis-witness and covolume-index argument. On each compact subset of lattice moduli space, the same argument yields a finite cutoff; a determinant-one cusp family shows that the basis-witness cutoff supplied by this method cannot be uniform on the full moduli space. Finally, a published rank-32 theta-series coincidence, together with a minimum-shell spanning identity, is stabilized by orthogonal direct sums to every rank n ≥ 32. If k(n) denotes the least degree of full Gram theta data that universally determines rank-n Euclidean lattices, we obtain 4 ≤ k(n) ≤ n for n ≥ 32. The paper separates proved statements, cited lattice-theoretic input, and computational checks; it does not claim a solution of the exact minimum-degree problem.

Primarily human-written textHuman understanding: all parts
Nov 04, 2026PDF ↗View work →

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