Inertia and Elementary Theories of Hermitian Nilpotent Operators

Let $V$ be a finite-dimensional Hermitian space over an algebraically closed field with a nonidentity involution, and suppose its form $h$ is nondegenerate. For a self-adjoint nilpotent operator $N$, we recover the multiplicity of each signed Jordan block from finite differences of the ranks and signature differences of the forms $h(x,N^j y)$. 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 $N$ 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 $R(\sqrt{-1})$ as over $\mathbb C$ for every real closed field $R$. The least Naimark dilation dimension of a measurement is the sum of its effect ranks.