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Inertia and Elementary Theories of Hermitian Nilpotent Operators

Primarily human-written textHuman understanding: some partsmath.LO — Logicmath.RA — Rings and Algebras

Contributed by Zhenyu Sun ↗, Jilin Wang ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

The manuscript is primarily human-written. OpenAI Codex assisted with editing the English prose, checking consistency between statements and proofs, and preparing the LaTeX source. The accompanying Python script uses exact rational arithmetic for the finite checks reported in the appendix. At the time of submission, parts of the manuscript have been checked and understood by a human.

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Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentNov 04, 2026