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  pdftitle={Inertia and Elementary Theories of Hermitian Nilpotent Operators},
  pdfauthor={Zhenyu Sun and Jilin Wang},
  pdfsubject={Hermitian algebra and model theory},
  pdfkeywords={Hermitian form, inertia, nilpotent operator, real closed field, model completeness}
}
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\title{Inertia and Elementary Theories\\of Hermitian Nilpotent Operators}
\author{Zhenyu Sun \and Jilin Wang}
\date{}

\begin{document}
\maketitle
\begin{abstract}
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.
\end{abstract}

\noindent\textbf{Keywords.}
Hermitian form, inertia, self-adjoint nilpotent operator, model completeness,
real closed field, quantum measurement.

\section{Introduction}\label{sec:introduction}

The ranks of the powers of a nilpotent operator determine the
lengths of its Jordan blocks. If the operator is self-adjoint for an
indefinite Hermitian form, each block also has a sign. Even on a
hyperbolic plane, two rank-one nilpotent operators can have the same
Jordan form and opposite signs. We ask how to read these signs from
the operator and the form, and which first-order sentences distinguish
the resulting structures.

Let \(K\) be an algebraically closed field with a nonidentity involution
\(\sigma\), and let \((V,h)\) be a nondegenerate Hermitian space of
dimension \(n\) over \(K\). The fixed field \(R=K^\sigma\) is real
closed, and \(K=R(u)\) with \(u^2=-1\). We write \((p,q)\) for the
ordered signature of \(h\). The four sorts of our language are the field
\(K\), the space \(V\), its full endomorphism algebra, and its
projective space. It names neither a basis of \(V\) nor a square root
of \(-1\).

\subsection{One operator and two operators}

For a self-adjoint nilpotent operator \(N\), consider the Hermitian forms
\[
 B_j^N(x,y)=h(x,N^j y),\qquad 0\leq j<n.
\]
Write \((p_j,q_j)\) for their positive and negative indices, and set
\(r_j=p_j+q_j\) and \(d_j=p_j-q_j\). The rank \(r_j\) equals
\(\rank N^j\). Second differences of the rank sequence give the number of blocks of each
length. The difference \(d_{m-1}-d_{m+1}\) gives the excess of positive
over negative blocks of length \(m\). Thus if \(c_m^\pm\) are the
multiplicities of the blocks with operator \(J_m(0)\) and Gram matrix
\(\pm F_m\), where \(F_m\) is the reversal matrix, then
\begin{equation}\label{eq:intro-inversion}
c_m^\pm=\frac12\left[
r_{m-1}-2r_m+r_{m+1}
\ \pm\ (d_{m-1}-d_{m+1})\right],
\qquad 1\leq m\leq n,
\end{equation}
with \(r_n=r_{n+1}=d_n=d_{n+1}=0\).
Theorem~\ref{thm:inertia-recovery} proves the formula, and
Corollary~\ref{cor:admissible-inertia} determines which sequences of
inertias occur. Each inertia is definable in the original language.
Consequently, a finite list of inertia conditions defines each
nilpotent unitary orbit.

Naming \(N\) as an operator constant gives one complete theory for
each admissible set of signed block multiplicities
(Theorem~\ref{thm:nilpotent-elementary-classification}). The reason
is that, in a signed Jordan basis, both the form and the operator have
integral matrices. Every sentence on that branch translates into an
ordered-field sentence with no scalar parameters. The translation
decides the sentence on each branch, and a canonical basis in a
smaller model remains a basis under any embedding. This gives
decidability and model completeness in the original language. Each
theory is NIP (it does not have the independence property).
Corollary~\ref{cor:branch-count} counts the completions at each
signature. When no operator is named there are exactly \(n+1\)
completions, one per signature. In both settings the fixed field is
stably embedded and has the induced structure of a pure real closed
ordered field with parameters.

For two operators, the complete theories can depend on a scalar
parameter. Fix \(1\leq s\leq\min(p,q)\), and let \(N_1,N_2\) be
self-adjoint square-zero operators with a common image \(W\) of
dimension \(s\). The forms \(h(x,N_i y)\) descend to nondegenerate
forms on \(V/W^\perp\). When both quotient forms are positive definite,
their ordered generalized eigenvalues classify the pair up to
simultaneous unitary conjugacy on a fixed Hermitian space
(Theorem~\ref{thm:pair-classification}). The inertia of
\[
 h\bigl(x,(N_2-tN_1)y\bigr),\qquad t\in R,
\]
recovers each eigenvalue and its multiplicity. The
ranks and inertias of all operator words together can be constant for
pairs with distinct spectra (Example~\ref{ex:word-inertia}).

Two such pairs are elementarily equivalent precisely when their ordered
spectra satisfy the same parameter-free ordered-field formulas. At
fixed \((p,q;s)\), the family theory is model complete, decidable,
and NIP, and has \(2^{\aleph_0}\) completions. The Turing degree of
each completion is the degree of its spectral type
(Theorem~\ref{thm:pair-elementary}).

\subsection{Relation to earlier work}

The classification of forms and self-adjoint operators under change of
basis is classical \cite{Scharlau1985,Sergeichuk1988,GohbergLancasterRodman2005}.
Over \(\mathbb C\), \citet[Theorem~6.4]{Mehl2006} gives the signed
Jordan blocks used here for the eigenvalue zero.
\citet[Section~6.1]{Pazzis2024} describes Hermitian invariants on
quotients of kernels. For a nilpotent self-adjoint operator, the
signature difference of the ambient form is the sum of the signs of
the odd blocks \cite[Lemma~1]{Mehl2020}. Applying this calculation
to each power form on \(V\) gives \eqref{eq:intro-inversion}, and
the resulting inertia predicates isolate the complete theories.

First-order descriptions of quantum systems depend on which objects
the language quantifies over. The lattice approach originates with
\citet{BirkhoffVonNeumann1936}. For fixed dimension, see
\citet{DunnHaggeMossWang2005} on quantum logics and
\citet[Theorem~2.5]{DunnMossWang2013} on uniform decidability of
the corresponding first-order lattice theories.
\citet[Theorem~6.2]{XuBartheZhou2025} reduce a typed Dirac language
in fixed finite dimensions to real closed fields by translating
quantification over scalars, vectors, dual vectors, and operators
into coordinates. Our translation also uses coordinates. For a
distinguished nilpotent it uses the integral signed Jordan matrices,
and for a positive pair it retains the ordered spectrum as field
parameters. In a different language, with systems as quantified objects,
\citet[Theorem~14]{Brunet2026} proves completeness of a possibilistic
theory for each dimension \(d\geq3\). In the analytic setting,
\citet{ArgotyPulido2014} studies Hilbert spaces with a distinguished
unbounded closed self-adjoint operator.

The role of the scalar field in quantum theory is discussed by
\citet{GudderPiron1971,CassinelliLahti2017}. Algebraic approaches
over other scalar fields include
\citet{SchumacherWestmoreland2012,Thas2020}.
\citet[Sections~3.3 and~4.2]{Thas2023} describes the real closed
fixed field and proposes a quantum Lefschetz principle. When no operator is named, two structures in our class satisfy the same
sentences precisely when their signatures agree.

Sections~\ref{sec:structures} and~\ref{sec:interpretations} develop
the inertia predicates and the coordinate translation used in the
classifications of Sections~\ref{sec:nilpotent}--\ref{sec:pairs}.
Section~\ref{sec:spectral} gives a square-zero criterion for the Witt
index over any field with involution and characterizes the existence
of orthogonal eigenbases over algebraically closed fields.
Section~\ref{sec:measurements} proves transfer for finite measurements
in the positive definite case and computes the least Naimark dilation
dimension. Appendix~\ref{app:formulas} gives first-order formulas
and finite counts by signature.
\section{Hermitian structures and their first-order language}
\label{sec:structures}

Throughout, \(n\geq1\) is a fixed integer. A field with involution is a field
\(K\) with an automorphism \(\sigma\) satisfying \(\sigma^2=\Id\).
The involution is allowed to be the identity in the structural definitions.
A Hermitian form is conjugate-linear in its first argument and linear in
its second, with
\[
h(ax,by)=\sigma(a)b\,h(x,y),
\qquad h(y,x)=\sigma(h(x,y)).
\]
All ambient forms \(h\) are nondegenerate. A subspace \(W\) is totally
isotropic when \(h(W,W)=0\). We define the Witt index \(w(h)\) as the
maximum dimension of such a subspace. This definition applies also when
the characteristic is \(2\).

\subsection{The scalar fields and inertia}

\begin{lemma}[Artin--Schreier]\label{lem:real-closed-field}
If \(K\) is algebraically closed and \(\sigma\neq\Id\), then
\(\operatorname{char}K=0\), the fixed field \(R=\Fix(\sigma)\) is real closed,
and
\[
K=R(u),\qquad u^2=-1,\qquad \sigma(u)=-u.
\]
Every \(z\in K\) has a unique expression \(a+ub\) with \(a,b\in R\).
Conversely, the complexification of any real closed field, equipped with
its conjugation, has these properties.
\end{lemma}

\begin{proof}
The fixed-field theorem gives \([K:R]=2\). The Artin--Schreier theorem on
fields with a proper finite algebraic closure implies that \(R\) is real
closed of characteristic zero and that its algebraic closure is \(R(u)\)
with \(u^2=-1\). The nonidentity \(R\)-automorphism sends \(u\) to \(-u\).
The converse is the usual characterization of real closed fields
\cite{ArtinSchreier1927,ArtinSchreier1927b}.
\end{proof}

In this situation the order on \(R\) is unique and definable. We use the
field formulas
\begin{align}
\operatorname{Real}(a)&:\ \sigma(a)=a, \label{eq:real}\\
\Pos(a)&:\ \operatorname{Real}(a)\ \wedge\
 \exists b\,(\sigma(b)=b\wedge b\neq0\wedge b^2=a).
 \label{eq:positive}
\end{align}
The relations \(a>b\) and \(a\geq b\), for \(a,b\in R\), abbreviate
\(\Pos(a-b)\) and \(\Pos(a-b)\vee a=b\).
If \(z=a+ub\), then
\[
\sigma(z)z=a^2+b^2,
\]
which is positive for \(z\neq0\).

For a possibly degenerate Hermitian form \(b\), its inertia is the triple
\(\In(b)=(p(b),q(b),z(b))\) of positive, negative, and zero indices.
We often write only the positive and negative indices, since
\(z(b)=n-p(b)-q(b)\) for a form on \(V\). The rank is \(p(b)+q(b)\),
and the signature difference is \(p(b)-q(b)\).

\begin{proposition}\label{prop:inertia}
Over \(K=R(u)\), every Hermitian form is congruent to
\(\diag(I_p,-I_q,0_z)\), and \((p,q,z)\) is unique.
For a nondegenerate form of signature \((p,q)\),
\[
w(h)=\min(p,q).
\]
\end{proposition}

\begin{proof}
If a Hermitian form \(b\) is nonzero, polarization shows that
\(b(v,v)\neq0\) for some \(v\). Indeed, the equalities
\(b(x+y,x+y)=0\) and \(b(x+uy,x+uy)=0\), together with vanishing diagonal
values, force both the real and imaginary parts of \(b(x,y)\) to vanish.
Split off the nondegenerate line \(Kv\) and proceed by induction.
Each nonzero diagonal entry lies in \(R\) and can be rescaled to \(1\)
or \(-1\). The remaining subspace is the radical.

In the resulting diagonal representation, projection of a positive
definite subspace onto the \(p\) positive coordinates is injective.
Thus its dimension is at most \(p\), and the coordinate subspace attains
this bound. It follows that \(p\) is intrinsic. The same argument
applies to \(q\), and the radical determines \(z\).

For \(h=\diag(I_p,-I_q)\), the vectors \(e_i+f_i\), for
\(1\leq i\leq\min(p,q)\), span a totally isotropic subspace.
Projection of any totally isotropic subspace onto either the positive or
the negative coordinates is injective, proving the reverse bound.
These are the usual Hermitian inertia and Witt arguments
\cite{Scharlau1985}.
\end{proof}

We write \(H_{p,q}=\diag(I_p,-I_q)\) for the standard matrix of
signature \((p,q)\).

\subsection{The four sorts}

The language \(\LQ\) has sorts \(K,V,M,P\). The scalar sort carries
\(0,1,+,-,\cdot,\sigma\). The vector and operator sorts carry their
additive and scalar-multiplication operations. The language also includes the maps
\[
M\times M\longrightarrow M,\quad (S,T)\longmapsto ST,
\qquad
\ev:M\times V\longrightarrow V,
\qquad h:V\times V\longrightarrow K,
\]
and a relation \(\Rep\subseteq V\times P\). We write \(Tv\) for
\(\ev(T,v)\). No basis is named in \(\LQ\).

\begin{definition}\label{def:structure}
The Hermitian structure associated with \((K,\sigma,V,h)\) is
\[
\Qstr(K,V,h)
=\bigl(K,V,\End_K(V),\bP(V);\,\sigma,h,\ev,\Rep\bigr),
\]
with the natural operations, where \(\dim_KV=n\) and
\[
\Rep(v,p)\quad\Longleftrightarrow\quad v\neq0,\ p=Kv.
\]
The theory \(\Tstr\) describes these structures over arbitrary fields with
involution. The theory \(\Tinv\) adds algebraic closedness of \(K\) and
\(\sigma\neq\Id\). Its signature-\((p,q)\) extension, for \(p+q=n\), is
denoted \(T_n^{p,q}\).
\end{definition}

\begin{proposition}\label{prop:axiomatization}
The class in Definition~\ref{def:structure} is elementary. In particular,
\(\Tstr\) has an effective finite axiomatization.
\end{proposition}

\begin{proof}
Use the field and involution axioms, the vector-space axioms, and the
Hermitian identities and nondegeneracy. The assertion that \(V\) has a
basis of length \(n\) is first-order. On \(M\), impose the vector-space
axioms, linearity of evaluation in its vector argument, and
\[
(S+T)v=Sv+Tv,\qquad (aT)v=a(Tv),\qquad
(-_M T)v=-_V(Tv),\qquad (ST)v=S(Tv).
\]
Add extensionality,
\[
\forall S\,\forall T\,
  \bigl((\forall v\ Sv=Tv)\longrightarrow S=T\bigr),
\]
and the assertion that for some basis \(e_1,\ldots,e_n\), every list of
images \(v_1,\ldots,v_n\) is realized by an operator, as expressed by
\[
\exists\bar e\left(\operatorname{Basis}_n(\bar e)\wedge
 \forall v_1\cdots v_n\ \exists T\ \bigwedge_i Te_i=v_i\right).
\]
These axioms identify \(M\) with all of \(\End_K(V)\).

For \(P\), require each nonzero vector to represent a unique point,
every point to have a representative, no zero vector to represent a
point, and
\[
\Rep(v,p)\wedge a\neq0\ \longrightarrow\ \Rep(av,p),
\]
\[
\Rep(v,p)\wedge\Rep(w,p)\
\longrightarrow\ \exists a\,(a\neq0\wedge w=av).
\]
Thus the fibers are exactly the nonzero parts of the one-dimensional
subspaces. For fixed \(n\) these conditions are a finite, effectively given list of
axioms. Evaluation identifies the operator sort with \(\End_K(V)\),
and the representative relation identifies the point sort with
\(\mathbf P(V)\).
\end{proof}

We use \(I\) for the unique operator satisfying \(\forall v\ Iv=v\). For each \(T\in M\), the
nondegeneracy of \(h\) gives a unique adjoint satisfying
\[
h(Tx,y)=h(x,T^\dagger y).
\]
In a basis with Gram matrix \(H\), it is
\[
T^\dagger=H^{-1}\sigma(T)^{\mathrm t}H.
\]
Consequently the relations \(S=T^\dagger\), \(T=T^\dagger\), and
\(T^\dagger T=TT^\dagger=I\) are first-order. They define adjunction,
self-adjointness, and unitarity, respectively.

\subsection{Inertia as an intrinsic first-order predicate}

Let \(g(x,y;\bar c)\) be a term defining a Hermitian form on \(V\).
For nonnegative integers \(a,b\) with \(a+b\leq n\), let
\(\operatorname{In}_{a,b}(g)\) assert that some basis
\(v_1,\ldots,v_n\) has Gram matrix
\[
\diag(I_a,-I_b,0_{n-a-b})
\]
for \(g\). The basis condition itself has an existential definition
over \(\Tstr\). It asks for vectors \(w_1,\ldots,w_n\) satisfying
\[
h(w_i,v_j)=\delta_{ij}\qquad(1\leq i,j\leq n).
\]
Pairing a linear relation among the \(v_j\) with each \(w_i\) shows
that all its coefficients vanish. Conversely, nondegeneracy of \(h\)
gives such a dual basis for every basis of \(V\).

\begin{lemma}\label{lem:inertia-definable}
In \(\Tinv\), the formula \(\operatorname{In}_{a,b}(g)\) defines exactly
the forms of inertia \((a,b,n-a-b)\). It has an existential definition
using vector and scalar variables. In particular the signature sentences
\(\operatorname{Sig}_{p,q}:=\operatorname{In}_{p,q}(h)\) are existential,
mutually exclusive, and exhaustive.
\end{lemma}

\begin{proof}
Proposition~\ref{prop:inertia} identifies the three indices from the Gram
matrix. The dual-basis condition above uses only existential quantifiers,
and the prescribed entries are the scalar terms \(0,1,-1\).
\end{proof}

When a fixed finite-dimensional quotient is used below, its linear
independence, basis, and form conditions can likewise be expressed on
representatives. Appendix~\ref{app:formulas} records the translations.
\section{Coordinate interpretations and semialgebraic reduction}
\label{sec:interpretations}

An orthogonal coordinate system need not be definable from the form.
Once a frame and a square root of \(-1\) are named, all four sorts
have coordinates in the ordered fixed field. For a named operator,
the only additional data needed are its matrix entries in that frame.

\subsection{Frames and projective representatives}

Fix a symmetric matrix \(H\in GL_n(\mathbb Z)\). Its entries are scalar
terms, and its inverse also has integer entries. An \(H\)-frame is a basis
\(\bar e=(e_1,\ldots,e_n)\) satisfying \(h(e_i,e_j)=H_{ij}\).
In the language expanded by \(\bar e\), the coordinate relations are
\begin{align}
v=\sum_i x_i e_i,\qquad
Te_j=\sum_i a_{ij}e_i. \label{eq:coordinates}
\end{align}
Their solutions are unique. In fact vector coordinates are given by
terms,
\[
x_i=\sum_j(H^{-1})_{ij}h(e_j,v),
\]
and the same expression applied to \(Te_j\) gives the operator entries.

For projective points, put
\[
C_i=\{(x_1,\ldots,x_n)\in K^n:
       x_1=\cdots=x_{i-1}=0,\ x_i=1\},
\qquad
C^{\mathrm{proj}}=\bigsqcup_{i=1}^n C_i.
\]
Each one-dimensional subspace has a unique representative in
\(C^{\mathrm{proj}}\), obtained by dividing by the first nonzero coordinate.
The formula
\[
\operatorname{Can}(p,\bar x):\
\bar x\in C^{\mathrm{proj}}\ \wedge\
\Rep\left(\sum_i x_i e_i,p\right)
\]
therefore specifies a definable bijection between \(P\) and this
definable subset of \(K^n\).

\begin{theorem}[Coordinate reduction]\label{thm:coordinate-reduction}
Expand a model of \(\Tinv\) by an \(H\)-frame and by an element \(u\)
satisfying \(u^2=-1\) and \(\sigma(u)=-u\).
Every formula in this expanded language admits an effective translation
to an ordered-field formula over \(R=\Fix(\sigma)\). The translation uses
two real coordinates for each scalar variable, \(2n\) for each vector
or projective variable, and \(2n^2\) for each operator variable.

With both the frame and \(u\) named, the structure is parameter-free
bi-interpretable with \(R\). If finitely many operators are also named,
the corresponding expansion of \(R\) names the real and imaginary
parts of their matrix entries. The same assertion holds uniformly when
those matrices are specified by polynomial or rational expressions in
real parameters named on both sides, with nonzero denominators.
\end{theorem}

\begin{proof}
First replace \(V\), \(M\), and \(P\) by \(K^n\), \(K^{n^2}\), and
\(C^{\mathrm{proj}}\), respectively, using \eqref{eq:coordinates}.
Vector operations, matrix multiplication, and evaluation are polynomial.
The form is
\[
h(x,y)=\sigma(x)^{\mathrm t}Hy.
\]
The representative relation becomes proportionality between a nonzero
vector and a canonical representative. Restricted projective
quantifiers are handled by the defining formula for
\(C^{\mathrm{proj}}\).

Next write each scalar \(x\) as \(a+ub\), where \(a,b\in R\), using
\[
a=\frac{x+\sigma(x)}2,\qquad
b=-\frac{u(x-\sigma(x))}2.
\]
Multiplication and conjugation become
\[
(a,b)(c,d)=(ac-bd,ad+bc),\qquad
\sigma(a,b)=(a,-b).
\]
Translate atomic formulas by these rules and proceed through Boolean
operations and quantifiers. This gives the claimed effective
translation and coordinate counts.

Conversely, the real field is interpreted on the definable fixed-field
subset of \(K\), with the order from \eqref{eq:positive}. Starting from
an ordered real closed field, form \(R^2\) with the displayed operations
and \(u=(0,1)\), then the vector, operator, and canonical projective
sorts. The coordinate maps and the scalar decomposition give definable
isomorphisms between each structure and the corresponding double
interpretation. Named operator matrices are transported by the same
maps, including in the parameterized case.
\end{proof}

Real closed fields have effective quantifier elimination in the ordered
field language \cite{Marker2002,vandenDries1998}. Hence every formula in
Theorem~\ref{thm:coordinate-reduction} has a semialgebraic description in
the resulting real coordinates. If an operator is fixed in a canonical
matrix form with entries in \(\{0,1,-1\}\), it introduces no scalar
parameters. In particular, the signed Jordan matrices used in
Section~\ref{sec:model-theory} require no expansion of the fixed field.

\subsection{Descent and the original language}

\begin{proposition}\label{prop:frame-descent}
Work in the class of models admitting an \(H\)-frame. Let
\(\varphi(\bar x;\bar e)\) be a formula using such a frame. Suppose
its truth on a tuple \(\bar x\) is independent of the chosen \(H\)-frame.
Then the relation it defines is definable in the original language by
\[
\exists\bar e\,
\bigl(\Frame_H(\bar e)\wedge\varphi(\bar x;\bar e)\bigr).
\]
\end{proposition}

\begin{proof}
An \(H\)-frame exists in every structure under consideration. The
assumed independence makes the existential choice equivalent to any
particular choice.
\end{proof}

The inertia predicates of Lemma~\ref{lem:inertia-definable} are already
formulas of \(\LQ\). Proposition~\ref{prop:no-quantifier-elimination}
and Remark~\ref{rem:frame-and-biinterpretation} describe which
conclusions about coordinates depend on the named constants.
\section{Inertia of powers and nilpotent unitary orbits}\label{sec:nilpotent}

Let $K$ be an algebraically closed field with a nonidentity involution
$\sigma$, let $R=K^\sigma$, and let $(V,h)$ be a nondegenerate Hermitian
space of dimension $n$ and signature $(p,q)$.  Throughout this section,
$N$ is an $h$-self-adjoint nilpotent endomorphism of $V$.  For
$0\leq j\leq n$, define the Hermitian form on $V$ by
\begin{equation}\label{eq:power-forms}
 B_j^N(x,y)=h(x,N^j y),\qquad B_0^N=h.
\end{equation}
Write $(p_j,q_j)$ for its positive and negative indices, and put
\begin{equation}\label{eq:power-inertia-data}
 r_j=p_j+q_j,\qquad d_j=p_j-q_j.
\end{equation}
The zero index is $n-r_j$.  Self-adjointness and nondegeneracy give
\begin{equation}\label{eq:power-radical}
 \operatorname{rad} B_j^N=\ker N^j,\qquad
 r_j=\operatorname{rank}N^j.
\end{equation}
In particular, $r_0=n$, $d_0=p-q$, and $r_n=d_n=0$.
We also set $r_{n+1}=d_{n+1}=0$.

The signed Jordan form over the complex numbers is classical
\cite{Mehl2006}. We construct its nilpotent blocks over \(K=R(u)\),
then compute the contribution of one block to each \(B_j^N\).
This calculation recovers the signs in any basis.

\subsection{Signed cyclic decompositions}\label{subsec:signed-cyclic}

Let $J_m=J_m(0)$ be the nilpotent Jordan matrix with ones immediately
above the diagonal, and let
\[
 F_m=(\delta_{a+b,m+1})_{1\leq a,b\leq m}
\]
be the reversal matrix.  Thus $F_mJ_m=J_m^{\mathrm t}F_m$.
The inertia of $F_m$ is
$(\lceil m/2\rceil,\lfloor m/2\rfloor)$. Each pair of coordinate
vectors interchanged by $F_m$ contributes one positive and one negative
square, and the middle coordinate contributes a positive square when
$m$ is odd.

\begin{lemma}\label{lem:signed-cyclic}
Suppose that $N^m=0$ and $N^{m-1}\neq0$.  There is an
$N$-invariant nondegenerate subspace $W$ of dimension $m$ and a basis
of $W$ in which
\[
 [N|_W]=J_m,\qquad [h|_W]=\varepsilon F_m,
 \qquad \varepsilon\in\{1,-1\}.
\]
The orthogonal complement $W^\perp$ is also $N$-invariant.
\end{lemma}

\begin{proof}
Since $N^{m-1}\ne0$ and $h$ is nondegenerate, $B_{m-1}^N$ is nonzero.
By Proposition~\ref{prop:inertia} it has a nonzero diagonal value.
Choose $v$ with

\[
 c=h(v,N^{m-1}v)\neq0.
\]
Every moment $\beta_j=h(v,N^jv)$ belongs to $R$, because $N^j$ is
self-adjoint.  Replacing $v$ by $|c|^{-1/2}v$ gives
$\beta_{m-1}=\varepsilon\in\{1,-1\}$.  The vectors
$v,Nv,\ldots,N^{m-1}v$ are linearly independent.

To make the Gram matrix a signed reversal matrix, we change the cyclic
generator so that its moments of orders $0,\ldots,m-2$ vanish.  In the truncated polynomial ring $R[t]/(t^m)$, put
\[
 C(t)=\varepsilon^{-1}
       \sum_{a=0}^{m-1}\beta_{m-1-a}t^a.
\]
Its constant term is $1$.  Consequently there is a polynomial $P(t)$
with constant term $1$ such that
\begin{equation}\label{eq:moment-normalization}
 P(t)^2C(t)\equiv1\pmod{t^m}.
\end{equation}
For example, take the truncated binomial expansion
\[
 P(t)=\sum_{\ell=0}^{m-1}
       \binom{-1/2}{\ell}\bigl(C(t)-1\bigr)^\ell
       \pmod{t^m}.
\]
All its coefficients lie in $R$.  Set $w=P(N)v$.
Since $P(N)$ commutes with $N$, is self-adjoint, and is invertible on
$V$, the vector $w$ generates the same cyclic subspace as $v$.
Writing $P(t)^2=\sum_a\gamma_at^a$ modulo $t^m$ gives
\[
 h(w,N^jw)
 =\sum_{a=0}^{m-1-j}\gamma_a\beta_{j+a}
 =\varepsilon\,[t^{m-1-j}]\bigl(P(t)^2C(t)\bigr)
 =\varepsilon\,\delta_{j,m-1}
 \quad(0\leq j<m).
\]
Here $[t^a]$ denotes the coefficient of $t^a$.

For $1\leq a\leq m$, put $e_a=N^{m-a}w$.  Then
$Ne_1=0$ and $Ne_a=e_{a-1}$ for $a>1$.  Moreover,
\[
 h(e_a,e_b)=h(w,N^{2m-a-b}w)
           =\varepsilon\,\delta_{a+b,m+1},
\]
where moments of index at least $m$ vanish.  This proves the two matrix
identities and nondegeneracy of the cyclic subspace $W$.
Finally, if $x\in W^\perp$ and $y\in W$, then
$h(Nx,y)=h(x,Ny)=0$, so $Nx\in W^\perp$.
\end{proof}

\begin{theorem}[Signed Jordan decomposition]\label{thm:signed-jordan}
There is an orthogonal decomposition of $(V,h,N)$ into blocks
\begin{equation}\label{eq:signed-blocks}
 (K^m,\,\varepsilon F_m,\,J_m),\qquad
 1\leq m\leq n,\quad \varepsilon\in\{1,-1\}.
\end{equation}
If $c_m^+$ and $c_m^-$ count the blocks of length $m$ with the indicated
sign, respectively, then
\begin{align}
 n&=\sum_{m=1}^n m(c_m^++c_m^-),\label{eq:signed-dimension}\\
 p&=\sum_{m=1}^n
 \left(\left\lceil\frac m2\right\rceil c_m^+
       +\left\lfloor\frac m2\right\rfloor c_m^-\right),\label{eq:signed-positive}\\
 q&=\sum_{m=1}^n
 \left(\left\lfloor\frac m2\right\rfloor c_m^+
       +\left\lceil\frac m2\right\rceil c_m^-\right).\label{eq:signed-negative}
\end{align}
Conversely, any nonnegative integral block multiplicities satisfying
\eqref{eq:signed-positive}--\eqref{eq:signed-negative} are realized by
an $h$-self-adjoint nilpotent endomorphism of $V$.
\end{theorem}

\begin{proof}
Apply Lemma~\ref{lem:signed-cyclic} with $m$ equal to the nilpotency
index, then repeat on the nondegenerate invariant orthogonal complement.
The process terminates because the dimension decreases at each step.
The signature formulas follow from the inertia of $F_m$.
For the converse, form the indicated orthogonal direct sum.  Its form
has signature $(p,q)$, so an isometry to $(V,h)$ transports its direct
sum of Jordan blocks to the required endomorphism.
\end{proof}

\subsection{Recovery and realization of inertia profiles}\label{subsec:inertia-recovery}

On a single signed block, taking a power of $N$ shortens the
nondegenerate part of the form by one coordinate at each step. Its
rank records the remaining length. Its signature difference records
the sign whenever that length is odd.

\begin{theorem}[Recovery from the inertia of powers]\label{thm:inertia-recovery}
For a self-adjoint nilpotent endomorphism $N$ with signed block
multiplicities $c_m^\pm$, one has
\begin{align}
 r_j&=\sum_{m>j}(m-j)(c_m^++c_m^-),\label{eq:rank-from-blocks}\\
 d_j&=\sum_{\substack{m>j\\m-j\text{ odd}}}
                  (c_m^+-c_m^-)
       \qquad(0\leq j\leq n).\label{eq:imbalance-from-blocks}
\end{align}
These relations are invertible.  More precisely, for $1\leq m\leq n$,
\begin{equation}\label{eq:inertia-inversion}
 c_m^\pm=\frac12\left[
 r_{m-1}-2r_m+r_{m+1}
 \ \pm\ (d_{m-1}-d_{m+1})\right].
\end{equation}
Consequently the sequence
$\bigl((p_j,q_j)\bigr)_{j=0}^{n-1}$ determines the signed Jordan
multiplicities uniquely.
\end{theorem}

\begin{proof}
On a block of length $m$ and sign $\varepsilon$, the Gram matrix of
$B_j^N$ is $\varepsilon F_mJ_m^j$.  If $j<m$, this matrix is
\begin{equation}\label{eq:shifted-reversal}
 \varepsilon F_mJ_m^j
   =0_j\oplus\varepsilon F_{m-j}.
\end{equation}
For $j\geq m$ it is zero.  Thus this block contributes $m-j$ to the
rank when $m>j$, and contributes $\varepsilon$ to the difference of
positive and negative indices exactly when $m-j$ is odd.
Additivity of inertia under orthogonal sums proves
\eqref{eq:rank-from-blocks}--\eqref{eq:imbalance-from-blocks}.

Set $t_m=c_m^++c_m^-$ and $e_m=c_m^+-c_m^-$.
Taking successive differences in \eqref{eq:rank-from-blocks} yields
\[
 r_{m-1}-r_m=\sum_{\ell\geq m}t_\ell,
 \qquad
 t_m=r_{m-1}-2r_m+r_{m+1}.
\]
Similarly, the summands in $d_{m+1}$ are exactly those in $d_{m-1}$
except for $e_m$, so
\[
 e_m=d_{m-1}-d_{m+1}.
\]
The equalities $c_m^\pm=(t_m\pm e_m)/2$ give
\eqref{eq:inertia-inversion}.  The specified zero boundary values
cover $m=n$ and $m=n-1$ as well.
\end{proof}

\begin{corollary}[Admissible inertia profiles]\label{cor:admissible-inertia}
Let $(p_j,q_j)$, $0\leq j<n$, be pairs of nonnegative integers with
$(p_0,q_0)=(p,q)$.  Put $r_j=p_j+q_j$, $d_j=p_j-q_j$ and append the
zero boundary values at $j=n,n+1$.  This sequence is the inertia
profile of an $h$-self-adjoint nilpotent endomorphism if and only if,
for every $1\leq m\leq n$, the integers
\[
 t_m=r_{m-1}-2r_m+r_{m+1},\qquad
 e_m=d_{m-1}-d_{m+1}
\]
satisfy
\begin{equation}\label{eq:profile-admissibility}
 t_m\geq0,\qquad |e_m|\leq t_m,
 \qquad t_m\equiv e_m\pmod2.
\end{equation}
\end{corollary}

\begin{proof}
Necessity follows from Theorem~\ref{thm:inertia-recovery}.
Conversely, \eqref{eq:profile-admissibility} makes
$c_m^\pm=(t_m\pm e_m)/2$ nonnegative integers.  The zero boundary values and telescoping give
\[
 r_j=\sum_{m>j}(m-j)t_m,\qquad
 d_j=\sum_{\substack{m>j\\m-j\text{ odd}}}e_m.
\]
At $j=0$ these identities show that the direct sum of the associated
signed blocks has dimension $n$ and signature $(p,q)$.
Theorem~\ref{thm:signed-jordan} realizes it on $(V,h)$.
The same identities at every $j$ recover the prescribed ranks and
index differences, hence the prescribed inertias.
\end{proof}

\begin{corollary}[Admissible Jordan partitions]\label{cor:admissible-partitions}
A partition $\lambda=(\lambda_1,\ldots,\lambda_a)$ of $n$ occurs as
the ordinary Jordan type of an $h$-self-adjoint nilpotent endomorphism
if and only if
\begin{equation}\label{eq:partition-admissibility}
 \sum_{i=1}^a\left\lfloor\frac{\lambda_i}{2}\right\rfloor
 \leq\min(p,q).
\end{equation}
Equivalently, the number $o(\lambda)$ of odd parts of $\lambda$
satisfies $o(\lambda)\geq|p-q|$.
\end{corollary}

\begin{proof}
Each block contributes at least $\lfloor\lambda_i/2\rfloor$ to both
indices.  This proves necessity.  Write
$b=\sum_i\lfloor\lambda_i/2\rfloor=(n-o(\lambda))/2$.
Every even block has equal positive and negative indices, whereas
each odd block contributes one additional index of the sign assigned
to that block.  If \eqref{eq:partition-admissibility} holds, assign
positive signs to $p-b$ odd blocks and negative signs to $q-b$ odd
blocks.  Both numbers are nonnegative and their sum is $o(\lambda)$.
The signs of even blocks may be chosen arbitrarily.  The resulting
signed direct sum has signature $(p,q)$.
\end{proof}

\subsection{First-order definitions of the unitary orbits}\label{subsec:nilpotent-definability}

Write $\operatorname{U}(h)$ for the group of invertible endomorphisms
preserving $h$.  Its action on endomorphisms is conjugation.

\begin{theorem}\label{thm:nilpotent-orbits}
Two $h$-self-adjoint nilpotent endomorphisms $N$ and $N'$ are conjugate
under $\operatorname{U}(h)$ if and only if
\[
 \operatorname{inertia}B_j^N=
 \operatorname{inertia}B_j^{N'}\qquad(0\leq j<n).
\]
Every such orbit is definable without parameters in the language $\LQ$, uniformly over the algebraically closed fields
with nonidentity involution and forms of signature $(p,q)$.
There are finitely many orbits, indexed by the nonnegative integral
solutions of \eqref{eq:signed-positive}--\eqref{eq:signed-negative}.
\end{theorem}

\begin{proof}
If $N'=gNg^{-1}$ with $g\in\operatorname{U}(h)$, then
\[
 B_j^{N'}(gx,gy)=h(gx,gN^jy)=B_j^N(x,y),
\]
so every inertia is preserved.  Conversely, equality of the inertias
and Theorem~\ref{thm:inertia-recovery} give equality of all signed
block multiplicities.  Match the signed block bases supplied by
Theorem~\ref{thm:signed-jordan}.  The resulting linear map preserves
$h$ and intertwines $N$ with $N'$, proving conjugacy.

For the definability assertion, apply Lemma~\ref{lem:inertia-definable}
to $B_j^T$. With $\Delta^{a,b}=\diag(I_a,-I_b,0_{n-a-b})$ the formula is
\begin{equation}\label{eq:inertia-formula}
 \operatorname{In}_{a,b}^{(j)}(T)\;\Longleftrightarrow\;
 \exists\bar v\left(
 \operatorname{Basis}_n(\bar v)\ \wedge \bigwedge_{\alpha,\beta=1}^n
 h(v_\alpha,T^jv_\beta)=\Delta^{a,b}_{\alpha\beta}\right).
\end{equation}
At $j=0$ we read $T^jv_\beta$ as $v_\beta$.

For an admissible profile $\mathcal I=((p_j,q_j))_{j=0}^{n-1}$, define
\begin{equation}\label{eq:nilpotent-orbit-formula}
 \Phi_{\mathcal I}(T):\quad
 \operatorname{Herm}(T)\ \wedge\ T^n=0_M\ \wedge\
 \bigwedge_{j=0}^{n-1}\operatorname{In}_{p_j,q_j}^{(j)}(T).
\end{equation}
Here $\operatorname{Herm}(T)$ abbreviates
$\forall x\,\forall y\,h(Tx,y)=h(x,Ty)$.
The first part of the theorem shows that \eqref{eq:nilpotent-orbit-formula}
defines exactly one orbit in every model of the fixed signature.
All coefficients in the formula are $0,1,-1$, and all tuple lengths
are fixed by $n$.  Finally, there are only finitely many possible
signed block multiplicities because their weighted sum is $n$.
\end{proof}

The disjunction of $\Phi_{\mathcal I}(T)\wedge\Phi_{\mathcal I}(S)$
over the profiles satisfying \eqref{eq:profile-admissibility} thus
defines unitary conjugacy on the nilpotent self-adjoint operators.

\begin{corollary}[Square-zero orbits]\label{cor:square-zero-orbits}
For $0\leq s\leq\min(p,q)$, there are exactly $s+1$ unitary orbits of
self-adjoint square-zero endomorphisms of rank $s$.  They are indexed
by
\[
 \operatorname{inertia}B_1^N=(a,s-a),\qquad 0\leq a\leq s.
\]
Their signed multiplicities are
\[
 c_2^+=a,\quad c_2^-=s-a,\quad
 c_1^+=p-s,\quad c_1^-=q-s.
\]
\end{corollary}

\begin{proof}
Square-zero operators have only Jordan blocks of sizes $1$ and $2$,
and rank $s$ means that exactly $s$ blocks have size $2$.
Each size-$2$ block contributes $(1,1)$ to the inertia of $h$,
so the size-$1$ multiplicities are as displayed.  Its contribution
to $B_1^N$ is a single nonzero square of its assigned sign.
Theorem~\ref{thm:nilpotent-orbits} now proves both existence and the
asserted count.
\end{proof}

\begin{example}\label{ex:rank-one-signs}
Take $H=F_2$ and $N_+=J_2$, $N_-=-J_2$.  Both operators are
$H$-self-adjoint, have rank $1$, and have ordinary Jordan type $(2)$.
Their first power forms have matrices
\[
 HN_+=\begin{pmatrix}0&0\\0&1\end{pmatrix},\qquad
 HN_-=\begin{pmatrix}0&0\\0&-1\end{pmatrix}.
\]
Their $B_1$-inertias are $(1,0)$ and $(0,1)$, respectively, so the
operators are not unitarily conjugate.  The rank sequences agree because they depend only on block lengths.
\end{example}

\begin{example}\label{ex:higher-power-signs}
On the same Hermitian space with Gram matrix $H=F_4\oplus F_2$,
consider
\[
 N_1=J_4\oplus(-J_2),\qquad N_2=(-J_4)\oplus J_2.
\]
The ambient signature is $(3,3)$.  Both operators have ordinary Jordan
type $(4,2)$ and rank sequence $(6,4,2,1,0,\ldots)$.  Their power-form inertias are
\[
\begin{array}{c|ccccc}
 j&0&1&2&3&j\geq4\\ \hline
 \operatorname{inertia}B_j^{N_1}&(3,3)&(2,2)&(1,1)&(1,0)&(0,0)\\
 \operatorname{inertia}B_j^{N_2}&(3,3)&(2,2)&(1,1)&(0,1)&(0,0)
\end{array}
\]
The first power that separates the orbits is $j=3$.  Formula~\eqref{eq:inertia-inversion} recovers a positive
length-$4$ block and a negative length-$2$ block for $N_1$, with both
signs reversed for $N_2$.
\end{example}

\subsection{The largest nilpotency index}\label{subsec:largest-nilpotency}

\begin{proposition}\label{prop:largest-nilpotency}
Let $w=\min(p,q)$.  The largest possible nilpotency index of an
$h$-self-adjoint nilpotent endomorphism is
\begin{equation}\label{eq:largest-nilpotency}
 \min(n,2w+1).
\end{equation}
The nilpotency index of the zero operator is taken to be $1$.
\end{proposition}

\begin{proof}
Suppose $N^m=0$ and $N^{m-1}\neq0$, and put $a=\lceil m/2\rceil$.
Then
\[
 h(N^ax,N^ay)=h(x,N^{2a}y)=0,
\]
so $\operatorname{im}N^a$ is totally isotropic.  A Jordan chain of
length $m$ gives
\[
 \dim\operatorname{im}N^a\geq m-a=\lfloor m/2\rfloor.
\]
Consequently $\lfloor m/2\rfloor\leq w$, whence $m\leq2w+1$.
The inequality $m\leq n$ is immediate.

If $p=q=w$, a single block $(F_{2w},J_{2w})$ attains the bound.
If $p>q$, use one block $(F_{2q+1},J_{2q+1})$ and $p-q-1$
positive length-$1$ blocks.  If $q>p$, use
$(-F_{2p+1},J_{2p+1})$ and $q-p-1$ negative length-$1$ blocks.
These constructions include $w=0$ and, by an isometry, give operators
on the prescribed space $(V,h)$.
\end{proof}
\section{Elementary classification and definable sets}
\label{sec:model-theory}

We use the original multisorted language $\LQ$, without a named basis or
imaginary unit, and the theories $\Tinv$ and $T_n^{p,q}$ from
Definition~\ref{def:structure}.  In the language $\LQ(N)$, where
$N$ is a constant of the operator sort, put
\[
 T_{n,\mathrm{nil}}
 =\Tinv+\{N=N^\dagger,\ N^n=0\}.
\]
Here the adjoint notation abbreviates its defining relation.  Adding the
signature sentence $\operatorname{Sig}_{p,q}$ gives
$T_{n,\mathrm{nil}}^{p,q}$, where $p+q=n$.

\subsection{Defining the signed Jordan branches}

Write
\[
 \Gamma_n=\left\{\gamma=(c_m^+,c_m^-)_{m=1}^n:
 c_m^\pm\in\mathbb Z_{\geq0},\quad
 \sum_{m=1}^n m(c_m^++c_m^-)=n\right\}.
\]
For $\gamma\in\Gamma_n$, form the canonical pair
$(J_\gamma,H_\gamma)$ by taking $c_m^+$ copies of $(J_m,F_m)$ and
$c_m^-$ copies of $(J_m,-F_m)$, in a fixed order, using the reversal matrix $F_m$ of Section~\ref{subsec:signed-cyclic}.
Thus $H_\gamma$ is symmetric, integral, and has determinant $\pm1$.
Let $(p_\gamma,q_\gamma)$ be its signature.  Define
\begin{align*}
 \operatorname{Can}_\gamma(\bar e,N):={}&
 \bigwedge_{a,b=1}^n h(e_a,e_b)=(H_\gamma)_{ab}
 \\
 &\wedge\bigwedge_{b=1}^n
 Ne_b=\sum_{a=1}^n(J_\gamma)_{ab}e_a,
 \qquad
 \chi_\gamma:=\exists\bar e\,
 \operatorname{Can}_\gamma(\bar e,N).
\end{align*}
The invertible Gram matrix makes the $n$ vectors in $\bar e$ a basis.  Set
\[
 T_{n,\mathrm{nil}}^{\gamma}=T_{n,\mathrm{nil}}+\{\chi_\gamma\}.
\]

Equivalently, one can specify a branch by the power forms
$B_j^N(x,y)=h(x,N^j y)$, using the inertia predicates from
Lemma~\ref{lem:inertia-definable}.

If $(p_j^\gamma,q_j^\gamma)$ are the inertias of
$H_\gamma J_\gamma^j$, Theorems~\ref{thm:signed-jordan}
and~\ref{thm:inertia-recovery} give
\begin{equation}\label{eq:inertia-branch}
 T_{n,\mathrm{nil}}\models
 \chi_\gamma\ \longleftrightarrow\
 \bigwedge_{j=0}^{n-1}
 \operatorname{In}_{p_j^\gamma,q_j^\gamma}(B_j^N).
\end{equation}
Thus $\chi_\gamma$ can be read either as a condition on a single
Jordan basis or as a list of basis-independent inertia conditions.

\begin{theorem}[Elementary classification of nilpotent expansions]
\label{thm:nilpotent-elementary-classification}
For fixed $n$, the following statements hold.
\begin{enumerate}
\item The theories $T_{n,\mathrm{nil}}^{\gamma}$, for
$\gamma\in\Gamma_n$, are exactly the complete extensions of
$T_{n,\mathrm{nil}}$.  Two models are elementarily equivalent if and only
if the inertias of $h(x,N^j y)$ agree for $0\leq j<n$.
\item Every $\LQ(N)$-sentence is equivalent modulo
$T_{n,\mathrm{nil}}$ to a disjunction of some of the sentences
$\chi_\gamma$.  The set of branches occurring in this disjunction can be
computed from the sentence.
\item Each branch, the full theory $T_{n,\mathrm{nil}}$, and each
fixed-signature theory $T_{n,\mathrm{nil}}^{p,q}$ are model complete,
decidable, and NIP in the original language $\LQ(N)$.
\item After naming a canonical frame and an element $u$ with
$u^2=-1$ and $\sigma(u)=-u$, each branch is parameter-free
bi-interpretable with $\mathrm{RCF}$.
\end{enumerate}
\end{theorem}

\begin{proof}
The signed decomposition in Theorem~\ref{thm:signed-jordan} provides a
$\gamma$ for each model. The inversion formula
\eqref{eq:inertia-inversion} makes it unique. Hence the sentences
$\chi_\gamma$ partition the class of models.  Every branch has a model over $\mathbb C$.
Fix $\gamma$ and name a canonical frame $\bar e$ and an anti-fixed
imaginary unit $u$.  Theorem~\ref{thm:coordinate-reduction} identifies the
resulting structure with its real-closed fixed field by mutually inverse
interpretations.  The additional constant $N$ has the fixed integral
matrix $J_\gamma$, so it introduces no scalar parameter into these
interpretations.  This proves (4). In particular, each sentence of $\LQ(N)$ has on this
branch a parameter-free ordered-field translation.  Completeness of $\mathrm{RCF}$
therefore proves completeness of the branch.  Together with
\eqref{eq:inertia-branch}, this proves (1).

There are finitely many elements of $\Gamma_n$, and they can be
enumerated effectively.  For a given sentence $\varphi$, translate it on
each branch and apply the decision procedure for real closed fields.
If $\Gamma_n(\varphi)$ denotes the branches on which the translation is
true, then
\[
 T_{n,\mathrm{nil}}\models
 \varphi\ \longleftrightarrow\
 \bigvee_{\gamma\in\Gamma_n(\varphi)}\chi_\gamma.
\]
The empty disjunction denotes falsity.  This proves (2), decidability of
each branch, and decidability of the full theory.  Restricting the enumeration to $\gamma$ of signature $(p,q)$ decides
the fixed-signature theory.

Model completeness requires compatibility with embeddings.  Let
$f:\mathcal A\longrightarrow\mathcal B$ be an embedding between arbitrary
models of $T_{n,\mathrm{nil}}$.  Choose a canonical frame
$\bar e$ in $\mathcal A$, of type $\gamma$, and an anti-fixed imaginary
unit $u\in K_{\mathcal A}$.  The embedding preserves the Gram and operator equations in $\mathcal B$.  The matrix $H_\gamma$ is
invertible, so $f(\bar e)$ is a basis of $V_{\mathcal B}$.  In
particular, both models have the same signed Jordan data.

The restriction of $f$ to the fixed fields is an ordered-field embedding
\[
 R_{\mathcal A}\longrightarrow R_{\mathcal B}.
\]
Indeed, positive elements of a real closed field are its nonzero squares.
This embedding is elementary by model completeness of $\mathrm{RCF}$.
Relative to $(\bar e,u)$ and $(f(\bar e),f(u))$, the coordinates of vectors,
operators, and canonical projective representatives commute with $f$.
The effective translation of Theorem~\ref{thm:coordinate-reduction} thus
gives, for every formula $\varphi(\bar x)$ and every tuple $\bar a$ from
$\mathcal A$,
\[
 \mathcal A\models\varphi(\bar a)
 \quad\Longleftrightarrow\quad
 \mathcal B\models\varphi(f(\bar a)).
\]
This proves model completeness in the original language, both on each
branch and for the full theory.  It also proves the assertion for any
union of branches, including a fixed ambient signature.

Each framed branch is interpretable in the o-minimal theory
$\mathrm{RCF}$. It is therefore NIP, as is its reduct.
For any fixed formula, choose an independence bound on each of the
finitely many branches.  The maximum of these bounds works for the full
theory and for every union of branches.  This completes (3).
\end{proof}

\begin{corollary}[The theory without a distinguished operator]
\label{cor:unframed-hermitian-classification}
The theory $\Tinv$ has exactly $n+1$ complete extensions,
namely $T_n^{p,n-p}$ for $0\leq p\leq n$.
Each extension, and the theory \(\Tinv\) itself, is model complete,
decidable, and NIP.
Every sentence is effectively equivalent, modulo $\Tinv$,
to a disjunction of signature sentences.  Consequently, two
models of $\Tinv$ are elementarily equivalent precisely when their
ordered signatures agree.
\end{corollary}

\begin{proof}
Expand each structure definitionally by $N=0$.  Its signed Jordan data
are $c_1^+=p$, $c_1^-=n-p$, with all remaining multiplicities zero.
The preceding theorem applies to these $n+1$ branches.  Their
signatures distinguish them first-order, and every nondegenerate form
has exactly one such signature.
\end{proof}

The signatures $(n,0)$ and $(0,n)$ both have Witt
index zero. The sentence $\exists v\,h(v,v)=1$ distinguishes
them.  In fixed dimension $n$, a Witt stratum of index $r<n/2$ contains
the two complete branches $(r,n-r)$ and $(n-r,r)$. For $n=2r$ these coincide.

\subsection{Counting complete branches at fixed signature}

Let $b_{p,q}$ be the number of complete extensions of
$T_{p+q,\mathrm{nil}}^{p,q}$, with $b_{0,0}=1$ for the purpose of the
following generating function.

\begin{corollary}\label{cor:branch-count}
As an identity of formal power series,
\begin{equation}\label{eq:branch-generating-function}
 \sum_{p,q\geq0}b_{p,q}x^p y^q
 =\prod_{a\geq0}
 \frac{1}{(1-x^{a+1}y^a)(1-x^a y^{a+1})}
 \prod_{a\geq1}\frac{1}{(1-(xy)^a)^2}.
\end{equation}
Thus $b_{p,q}$ is the coefficient of $x^p y^q$ on the right.  In
particular,
\[
 b_{n,0}=b_{0,n}=1,\qquad
 b_{1,1}=3,\qquad b_{2,1}=b_{1,2}=4,\qquad b_{2,2}=10.
\]
\end{corollary}

\begin{proof}
A positive signed block of odd size $2a+1$ contributes signature
$(a+1,a)$, and a negative signed block contributes $(a,a+1)$.
Arbitrary multiplicities give the two corresponding geometric-series
factors.  For even size $2a$, either sign contributes signature $(a,a)$,
but the two signs represent distinct signed Jordan blocks and have
independent multiplicities.  They give the squared factor.  Theorem
\ref{thm:nilpotent-elementary-classification} identifies precisely these
multisets with complete branches.  Each coefficient receives
contributions from finitely many block sizes, so the product is
well-defined formally.  The displayed values follow by extracting the
corresponding coefficients.
\end{proof}

For signature $(1,1)$, the three branches consist of $N=0$ and the two
nonzero rank-one square-zero operators with signed blocks
$(J_2,F_2)$ and $(J_2,-F_2)$.  Their distinguishing invariant is the
inertia of $B_1^N(x,y)=h(x,Ny)$, which is respectively $(0,0)$, $(1,0)$,
and $(0,1)$.  The two nonzero operators therefore have the same ordinary Jordan
partition and distinct elementary theories.

\subsection{Semialgebraic definable sets and the fixed field}

\begin{theorem}[Relative semialgebraic reduction]
\label{thm:relative-semialgebraic-reduction}
Fix a branch $\gamma$, a canonical frame, and an anti-fixed imaginary
unit.  Every definable set, with parameters, in a finite product of the
sorts of $\LQ$ is carried by the coordinate maps to a semialgebraic set
over the real-closed fixed field.  The passage from a formula to a
Boolean combination of polynomial equalities and strict inequalities
is effective.  The coefficients lie in the subfield of $R$ generated by the real and
imaginary coordinates of the parameters over $\mathbb Q$.

In every model of $T_{n,\mathrm{nil}}$, the fixed field is stably
embedded and its induced structure, with parameters, is precisely that
of a pure real closed ordered field.  Both assertions also hold for
$\Tinv$.
\end{theorem}

\begin{proof}
Using the chosen frame and imaginary unit, the sorts $K,V,M$ are
identified with $R^2,R^{2n},R^{2n^2}$, respectively.  The projective sort
is identified with the finite union of charts whose first nonzero
complex coordinate is $1$.  This is a semialgebraic subset of
$R^{2n}$.  All domains and primitive symbols have uniform
ordered-field definitions, and the matrix of $N$ is the fixed integral
matrix $J_\gamma$.  Theorem~\ref{thm:coordinate-reduction} therefore
translates any formula, together with its parameters, to an
ordered-field formula over the corresponding real-coordinate tuple.
Quantifier elimination for real closed fields yields the stated
semialgebraic description and is effective
\cite{Marker2002,vandenDries1998}.  Denominators introduced by the
real--imaginary decomposition can be cleared, so no extra coefficients
are needed.  Conversely, semialgebraic conditions in these coordinates
are definable in the framed structure, using the definable order on
$R$.

To prove stable embeddedness, let $X\subseteq R^d$ be defined in the
full structure by a formula with an arbitrary finite tuple of parameters
from any of the four sorts.  Choose a canonical frame and an imaginary unit, and apply
the preceding translation.  Since the free variables lie in $R$, their
imaginary coordinates vanish, and the resulting definition uses only
ordered-field operations and a finite tuple of parameters from $R$.
Thus every such subset of $R^d$ is
definable over $R$ in its pure ordered-field structure.  The reverse
inclusion follows because the ordering of $R$ is definable by nonzero
squares.  This proves both stable embeddedness and the description of
the induced structure.  Taking the definitional expansion $N=0$ gives
the result without a distinguished operator.
\end{proof}

In the ordered-field coordinates, positivity is atomic. In $\LQ$ it
requires a quantifier, even if the frame and imaginary unit are named.

\begin{proposition}\label{prop:no-quantifier-elimination}
No theory $T_{n,\mathrm{nil}}^{\gamma}$ eliminates quantifiers in
$\LQ(N)$, even after a canonical frame and an anti-fixed imaginary
unit have been named.  The same holds for
$T_n^{p,q}$ with or without these constants.
\end{proposition}

\begin{proof}
Consider the definable subset
\[
 R_{>0}=\{x:\sigma(x)=x\ \wedge\
 \exists z\,(\sigma(z)=z\wedge z\ne0\wedge z^2=x)\}.
\]
In the framed language, every atomic formula with one free scalar
variable, restricted to $x\in R$, becomes a finite system of
one-variable polynomial equalities over $R(u)$.  To see this, apply induction on terms. The field operations, vector operations,
evaluation, composition, and the Hermitian form are polynomial in
coordinates, while the named frame and $N$ have fixed integral
coordinates.  There is no closed term of projective sort, so the
representative relation produces no additional atomic condition in
this one-variable situation.  Each such polynomial equality defines
a finite set or all of $R$.  Every quantifier-free formula therefore
defines a finite or cofinite subset of $R$.  The set $R_{>0}$ and its
complement in $R$ are both infinite, proving the assertion.  The same positive cone is definable in the reduct, whose quantifier-free
formulas are also quantifier-free in the expansion. Taking $N=0$ gives
the assertion for $T_n^{p,q}$.
\end{proof}

\begin{remark}\label{rem:frame-and-biinterpretation}
A frame chosen in the smaller model remains a frame under any
embedding, which is enough for model completeness. Bi-interpretability
imposes a further constraint because it preserves automorphism groups.
In an unframed model, $v\mapsto-v$ on $V$, acting identically on
$K,M,P$, is an automorphism of order two, also when $N$ is named. An
automorphism of an ordered field has no nontrivial finite orbit. Indeed,
if it moves an element upwards or downwards, its successive iterates
form a strictly monotone sequence. Thus the unframed model cannot be
parameter-free bi-interpretable with a real closed field. In the
standard complex model, even after a canonical frame is named, complex
conjugation in that frame gives an automorphism of order two.
Naming $u$ distinguishes it from $-u$, as required by the coordinate
interpretation.

\end{remark}
\section{Pairs of square-zero operators and Hermitian pencils}
\label{sec:pairs}

Two self-adjoint square-zero operators with the same image induce two forms on a
common quotient. When both forms are positive definite, their
generalized eigenvalues can vary even though the word ranks and
inertias remain fixed. Throughout this section, fix \(p+q=n\) and
\(1\leq s\leq\min(p,q)\).

\begin{definition}\label{def:positive-pair}
A positive pair of rank \(s\) consists of two named operators \(N_1,N_2\)
in a model of \(T_n^{p,q}\) such that
\[
N_i=N_i^\dagger,\qquad N_i^2=0,\qquad
\rank N_i=s,\qquad \im N_1=\im N_2,
\]
and
\begin{equation}\label{eq:positive-quotient-condition}
N_i x\neq0\quad\Longrightarrow\quad \Pos\bigl(h(x,N_i x)\bigr)
\qquad(i=1,2).
\end{equation}
The resulting first-order theory is denoted \(T_{\mathrm{pair}}^{p,q;s}\).
\end{definition}

Set \(W=\im N_1=\im N_2\). Self-adjointness gives
\(\ker N_i=(\im N_i)^\perp=W^\perp\), and \(N_i^2=0\) gives
\(W\subseteq W^\perp\). Thus
\[
N_1N_2=N_2N_1=0.
\]
On the \(s\)-dimensional quotient \(E=V/W^\perp\), define
\begin{equation}\label{eq:pair-quotient-forms}
q_i([x],[y])=h(x,N_i y).
\end{equation}
Changing either representative by an element of \(W^\perp\) does not
change this expression. The forms are Hermitian and nondegenerate.
Condition~\eqref{eq:positive-quotient-condition} says exactly that both
are positive definite.

\subsection{A simultaneous normal form}

\begin{theorem}\label{thm:pair-classification}
Every positive pair of rank \(s\) admits a basis in which
\begin{equation}\label{eq:pair-canonical}
H=
\begin{pmatrix}
0&I_s&0\\ I_s&0&0\\0&0&H_{p-s,q-s}
\end{pmatrix},
\quad
N_1=
\begin{pmatrix}
0&I_s&0\\0&0&0\\0&0&0
\end{pmatrix},
\quad
N_2=
\begin{pmatrix}
0&D_{\boldsymbol\lambda}&0\\0&0&0\\0&0&0
\end{pmatrix},
\end{equation}
where
\[
D_{\boldsymbol\lambda}=\diag(\lambda_1,\ldots,\lambda_s),
\qquad 0<\lambda_1\leq\cdots\leq\lambda_s,\quad \lambda_i\in R.
\]
The ordered tuple \(\boldsymbol\lambda\), with repetitions, is unique.
Over the same scalar field, it is a complete invariant for simultaneous
unitary conjugacy.
\end{theorem}

\begin{proof}
Let \(A\in\End_K(E)\) be defined by
\[
q_1(x,Ay)=q_2(x,y).
\]
The Hermitian symmetry of \(q_2\) makes \(A\) self-adjoint for \(q_1\).
Algebraic closedness gives an eigenvector \(v\ne0\). Its eigenvalue
is \(q_2(v,v)/q_1(v,v)>0\). The \(q_1\)-orthogonal complement of
\(Kv\) is invariant under \(A\).
Induction, with normalization in the real closed field, gives a
\(q_1\)-orthonormal eigenbasis with positive eigenvalues. Order them
as \(\lambda_1,\ldots,\lambda_s\).

Choose representatives \(f_i\in V\) of this basis and put \(e_i=N_1f_i\).
Then
\[
h(f_i,e_j)=\delta_{ij},\qquad
h(e_i,e_j)=0,\qquad
N_2f_j=\lambda_j e_j.
\]
Indeed, both sides of the last identity lie in \(W\) and have the
same pairings with every \(f_i\). These pairings determine a vector
in \(W\) because the \(e_j\) are a basis of \(W\) dual to the
\(f_i\) under \(h\).
Replace each \(f_j\) by
\[
f'_j=f_j-\frac12\sum_{\ell=1}^s e_\ell h(f_\ell,f_j).
\]
The pairings \(h(e_i,f'_j)=\delta_{ij}\) are unchanged, and direct
expansion gives \(h(f'_i,f'_j)=0\).
Since both operators kill \(W\), their values on the \(f'_j\) are also
unchanged. Thus the span of the \(e_i,f'_i\) is a nondegenerate
hyperbolic subspace. Its orthogonal complement has signature
\((p-s,q-s)\) and lies in \(W^\perp\), so both operators vanish there.
This proves \eqref{eq:pair-canonical}.

A simultaneous isometry induces an isometry of the pair \((q_1,q_2)\),
and hence preserves the spectrum of \(A=q_1^{-1}q_2\) with multiplicities.
Conversely, equality of the ordered spectra identifies the canonical
bases in \eqref{eq:pair-canonical} and produces a simultaneous isometry.
\end{proof}

The simultaneous reduction of two forms with one positive definite form
is the usual definite Hermitian pencil construction
\cite{HornJohnson2013,GohbergLancasterRodman2005}. Theorem
\ref{thm:pair-classification} realizes that construction inside the
square-zero operator geometry, including the common radical and the
ambient orthogonal complement.

\begin{proposition}\label{prop:pair-definable-spectrum}
The ordered spectrum \(\boldsymbol\lambda\) is a uniquely definable
tuple in the original language with \(N_1,N_2\).
For every \(t\in R\), the Hermitian pencil
\[
B_t(x,y)=h\bigl(x,(N_2-tN_1)y\bigr)
\]
has inertia
\begin{equation}\label{eq:pencil-inertia}
\In(B_t)=
\left(
\#\{i:\lambda_i>t\},
\#\{i:\lambda_i<t\},
n-s+\#\{i:\lambda_i=t\}
\right).
\end{equation}
\end{proposition}

\begin{proof}
The tuple is specified by \(0<\lambda_1\leq\cdots\leq\lambda_s\),
\(\sigma(\lambda_i)=\lambda_i\), and the existence of \(f_1,\ldots,f_s\)
such that
\begin{equation}\label{eq:spectrum-graph}
h(f_i,N_1f_j)=\delta_{ij},\qquad
N_2f_j=\lambda_jN_1f_j.
\end{equation}
The first condition makes their classes a basis of \(E\), since its
dimension is \(s\). The second condition specifies an eigenbasis for
\(A\). Theorem~\ref{thm:pair-classification} gives existence and uniqueness
of the ordered tuple. In that basis the quotient pencil is
\(\diag(\lambda_1-t,\ldots,\lambda_s-t)\). Pulling back to \(V\) contributes the \(n-s\) additional zero indices
from \(W^\perp\), giving \eqref{eq:pencil-inertia}. In particular,
the nullity at \(t\) minus \(n-s\) is the multiplicity of \(t\).
\end{proof}

\subsection{The complete theories of positive pairs}

For an ordered-field tuple \(\bar a\), the notation
\(\tp_{\RCF}(\bar a/\varnothing)\) means the set of parameter-free
ordered-field formulas satisfied by \(\bar a\) in its real closed field.

\begin{theorem}\label{thm:pair-elementary}
Let \(\mathcal A,\mathcal B\models T_{\mathrm{pair}}^{p,q;s}\), with
fixed fields \(R_A,R_B\) and ordered spectra
\(\boldsymbol\lambda_A,\boldsymbol\lambda_B\).
Then
\begin{equation}\label{eq:pair-elementary-equivalence}
\mathcal A\equiv\mathcal B
\quad\Longleftrightarrow\quad
\tp_{\RCF}(\boldsymbol\lambda_A/\varnothing)
=\tp_{\RCF}(\boldsymbol\lambda_B/\varnothing).
\end{equation}
The theory \(T_{\mathrm{pair}}^{p,q;s}\) is model complete, decidable,
and NIP. It has exactly \(2^{\aleph_0}\) completions.

Moreover, for each model its complete theory is Turing equivalent to
the real-closed-field type of its ordered spectrum.
\end{theorem}

\begin{proof}
For the forward implication in \eqref{eq:pair-elementary-equivalence},
interpret the ordered fixed field and use the uniquely definable tuple
from Proposition~\ref{prop:pair-definable-spectrum}. This transports any
ordered-field formula about the tuple to a sentence of the pair
language.

For the reverse implication, choose a canonical basis as in
\eqref{eq:pair-canonical} and name \(u\). Theorem
\ref{thm:coordinate-reduction} translates any pair-language sentence
\(\varphi\) into an ordered-field formula
\(\theta_\varphi(\boldsymbol\lambda)\), uniformly in the tuple.
Equal types give the same truth value to \(\theta_\varphi\), hence to
\(\varphi\).
Every complete type satisfying
\(0<\lambda_1\leq\cdots\leq\lambda_s\) is realized in some real closed
field, and \eqref{eq:pair-canonical} constructs a corresponding model.

To prove model completeness, consider an embedding
\(\mathcal A\subseteq\mathcal B\). Choose a canonical frame and \(u\)
in \(\mathcal A\). Its Gram matrix in \(\mathcal B\) is the same
invertible matrix, so the frame remains a basis over the larger field.
The two operator matrices remain exactly those in
\eqref{eq:pair-canonical}, so the embedding preserves the ordered spectrum. The inclusion \(R_A\subseteq R_B\) is
an embedding of real closed ordered fields, hence elementary, also
after naming that tuple. Coordinate reduction therefore makes the
original embedding elementary.

For decidability, a sentence \(\varphi\) is a consequence of the family
theory precisely when \(\RCF\) proves
\[
\forall\lambda_1\cdots\lambda_s\,
\bigl(0<\lambda_1\leq\cdots\leq\lambda_s
       \longrightarrow\theta_\varphi(\boldsymbol\lambda)\bigr).
\]
Quantifier elimination in \(\RCF\) decides this sentence. It also
suffices for the NIP assertion to consider the uniformly interpreted
framed family. In each translated formula, regard the spectral
coordinates as additional parameter variables. NIP of \(\RCF\) then
gives a bound for that formula independent of the choice of spectrum.
The bound still holds after the frame and \(u\) are forgotten.

Over \(\mathbb R\), take the largest eigenvalue in \((1,2)\) and every
other eigenvalue equal to \(1\). A rational number strictly between two
choices distinguishes their types. This gives \(2^{\aleph_0}\)
completions. The countable language gives the matching upper bound.

Finally the translations in both directions are effective. An oracle
for the real-field type decides \(\theta_\varphi\), hence every
sentence about the pair. Conversely, a formula about the ordered
spectrum is translated into a sentence using
\eqref{eq:spectrum-graph} and the interpreted fixed-field order.
These are the two claimed Turing reductions.
\end{proof}

An individual completion is decidable precisely when its spectral type
is decidable. For example, if all eigenvalues are real algebraic,
defining polynomials and rational isolating intervals specify the
ordered tuple, and quantifier elimination decides its type.

\begin{example}[Constant word inertias and varying pencil spectra]
\label{ex:word-inertia}
Take \(n=4\), \((p,q)=(2,2)\), \(s=2\), and use
\eqref{eq:pair-canonical} with
\[
D_t=\diag(1,t),\qquad t>1.
\]
Here an operator word means a monomial in \(N_1,N_2\).
Since \(t>1\), the operators \(N_1,N_2\) are linearly independent.
The empty word gives \(I\) and hence the ambient form \(h\).
Every operator word
of length at least two is zero. Both words of length one have rank
\(2\), and both forms \(h(x,N_i y)\) have inertia \((2,0,2)\).
Thus the ranks and inertias of all words in the two operators are
independent of \(t\).

The generalized spectrum is \((1,t)\). As \(t\) ranges over \((1,\infty)\subset\mathbb R\), the structures have pairwise different
complete theories by Theorem~\ref{thm:pair-elementary}. The second jump in the pencil inertia \eqref{eq:pencil-inertia}
occurs at \(t\).
\end{example}
\section{Witt index and orthogonal spectral decompositions}
\label{sec:spectral}

A square-zero self-adjoint operator has totally isotropic image.
Conversely, a basis of any totally isotropic subspace gives an explicit
operator with that image. Neither direction requires an assumption on
the scalar field or its characteristic.

\begin{theorem}\label{thm:general-witt}
Let \(h\) be a nondegenerate Hermitian form on a finite-dimensional
vector space over any field with involution. For each integer \(s\geq0\),
there exists a self-adjoint operator \(N\) with
\[
N^2=0,\qquad \rank N=s
\]
if and only if \(s\leq w(h)\).
Every totally isotropic \(s\)-dimensional subspace occurs as the image of
such an operator.
\end{theorem}

\begin{proof}
If \(N=N^\dagger\) and \(N^2=0\), then
\[
h(Nx,Ny)=h(x,N^2y)=0,
\]
so \(\im N\) is totally isotropic.

Conversely, let \(W=\Span(e_1,\ldots,e_s)\) be totally isotropic and put
\[
N(x)=\sum_{i=1}^s e_i h(e_i,x).
\]
This map is linear, and
\[
h(Nx,y)=\sum_i h(x,e_i)h(e_i,y)=h(x,Ny).
\]
Since \(h(e_i,e_j)=0\), one has \(N^2=0\).
To see that the image is all of \(W\), observe that a relation
\(\sum_i a_i h(e_i,x)=0\) for all \(x\) gives
\(h(\sum_i\sigma(a_i)e_i,x)=0\), hence all \(a_i=0\).
Thus the coordinate functionals \(h(e_i,\cdot)\) are linearly independent.
Their joint map has rank \(s\), so \(\im N=W\).
For \(s=0\), take \(N=0\).
\end{proof}

For fixed \(n\), this theorem gives first-order sentences
\[
\HSZ_s:\
\exists N\,\bigl(N=N^\dagger\wedge N^2=0\wedge\rank N=s\bigr).
\]
In \(T_n^{p,q}\), they hold precisely for \(s\leq\min(p,q)\).
Thus the entire list of square-zero ranks records the Witt index.
The proof applies in characteristic \(2\) as well, with the definition
of \(w(h)\) used in Section~\ref{sec:structures}.

\subsection{The spectral criterion}

An operator has a complete orthogonal eigenbasis if there are a basis
\(v_1,\ldots,v_n\) and scalars \(\lambda_1,\ldots,\lambda_n\) such that
\[
Tv_i=\lambda_i v_i,\qquad h(v_i,v_j)=0\quad(i\neq j).
\]
For fixed \(n\), this is a first-order predicate \(\operatorname{COEB}(T)\).

\begin{theorem}\label{thm:individual-spectral}
In a model of \(\Tinv\), a self-adjoint operator \(T\) has a complete
orthogonal eigenbasis if and only if it is diagonalizable and all its
eigenvalues belong to \(R=\Fix(\sigma)\).
\end{theorem}

\begin{proof}
For eigenvectors \(v,w\) with eigenvalues \(\lambda,\mu\),
self-adjointness gives
\[
\bigl(\sigma(\lambda)-\mu\bigr)h(v,w)=0.
\]
If an orthogonal eigenbasis exists, none of its vectors is isotropic,
since its Gram matrix is diagonal and nondegenerate. Setting \(v=w\)
therefore shows \(\lambda\in R\).

Conversely, suppose that \(T\) is diagonalizable and all its eigenvalues
are real. Distinct eigenspaces are orthogonal. Each eigenspace is
nondegenerate, because a vector in its radical would be orthogonal to
every eigenspace and hence to \(V\). Proposition~\ref{prop:inertia}
provides an orthogonal basis in each eigenspace. Their union is the
required basis.
\end{proof}

\begin{theorem}\label{thm:universal-spectral}
Over an algebraically closed field with any involution, every
self-adjoint operator on \((V,h)\) has a complete orthogonal eigenbasis
if and only if \(h\) is anisotropic. In the nonidentity-involution case,
this is equivalent to signature \((n,0)\) or \((0,n)\).
\end{theorem}

\begin{proof}
If \(h\) is isotropic, Theorem~\ref{thm:general-witt} gives a nonzero
self-adjoint square-zero operator. Such an operator is not diagonalizable.

Assume \(h\) is anisotropic and let \(T=T^\dagger\).
Algebraic closedness gives an eigenvector \(v\neq0\). The equality
\[
\sigma(\lambda)h(v,v)=h(Tv,v)=h(v,Tv)=\lambda h(v,v)
\]
shows \(\sigma(\lambda)=\lambda\). The line \(Kv\) is nondegenerate,
\(V=Kv\perp v^\perp\), and \(v^\perp\) is \(T\)-invariant. The
restricted form is again nondegenerate and anisotropic. Induction on
the dimension completes the proof. The last assertion follows from
Proposition~\ref{prop:inertia}.
\end{proof}

For example, over an algebraically closed field of characteristic
different from \(2\) with the identity involution, every nondegenerate
symmetric form of dimension at least \(2\) is isotropic. In each of these
dimensions a self-adjoint operator can therefore fail to have an
orthogonal eigenbasis.

\begin{example}\label{ex:nonreal-spectrum}
Over \(K=R(u)\) with its nonidentity involution, consider the hyperbolic
plane with Gram matrix
\(\begin{psmallmatrix}0&1\\1&0\end{psmallmatrix}\). The matrix
\(\begin{psmallmatrix}0&1\\0&0\end{psmallmatrix}\) is self-adjoint and
square-zero. A different obstruction occurs with
\[
H=\begin{pmatrix}1&0\\0&-1\end{pmatrix},
\qquad
T=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.
\]
Here \(T^\dagger=T\), \(T^2=-I\), and the eigenvalues are \(u,-u\).
Their eigenvectors \((1,u),(1,-u)\) are isotropic and have mutual
pairing \(2\). This operator is diagonalizable and has no orthogonal
eigenbasis.
\end{example}

\begin{remark}
By Lemma~\ref{lem:real-closed-field}, an algebraically closed field of
positive characteristic admits no nonidentity involution. There is
also a restriction inherent in Hermitian symmetry. Applying
\(h(y,x)=\sigma(h(x,y))\) twice shows that \(\sigma^2\) fixes every
value of \(h\). For \(v\ne0\), nondegeneracy makes the linear
functional \(h(v,\cdot)\) surjective onto \(K\), so \(\sigma^2=\Id\)
on all of \(K\).

\end{remark}
\section{Finite measurements over real closed fixed fields}
\label{sec:measurements}

Work in the positive definite theory \(T_n^{n,0}\), with fixed field
\(R\). The transition probability between the rays of two nonzero
vectors \(v,w\) is

\begin{equation}\label{eq:transition-probability}
\mathsf P([v],[w])
=\frac{h(v,w)h(w,v)}{h(v,v)h(w,w)}.
\end{equation}
The expression is independent of the representatives. Completing a
square,
\[
0\leq h\left(w-v\frac{h(v,w)}{h(v,v)},
             w-v\frac{h(v,w)}{h(v,v)}\right)
=h(w,w)-\frac{h(w,v)h(v,w)}{h(v,v)},
\]
shows \(0\leq\mathsf P\leq1\) in \(R\).
Its graph is first-order, after clearing the nonzero denominator.
The corresponding rank-one orthogonal projection is
\[
\Pi_{[v]}x=v\,\frac{h(v,x)}{h(v,v)}.
\]

\subsection{States and effects}

An operator \(A\) is positive semidefinite, written \(A\succeq0\), if
\[
A=A^\dagger,\qquad
\forall v\ \bigl(h(v,Av)=0\ \vee\ \Pos(h(v,Av))\bigr).
\]
A density operator satisfies \(\rho\succeq0\) and \(\tr\rho=1\).
An effect satisfies \(0\preceq E\preceq I\).
To define the trace of \(A\), quantify over a basis and require the
scalar in question to be the sum of the diagonal entries of \(A\) in
that basis. Invariance under change of basis gives a unique scalar.

Fix a positive integer \(m\). An \(m\)-outcome measurement is a tuple
\(E_1,\ldots,E_m\) with
\begin{equation}\label{eq:povm}
E_i\succeq0,\qquad \sum_{i=1}^m E_i=I.
\end{equation}
These are the finite-dimensional positive-operator-valued measurements
(POVMs). For a density operator \(\rho\), the outcome probabilities are
\[
\pi_i=\tr(\rho E_i).
\]
They lie in \(R_{\geq0}\) and sum to \(1\). To see nonnegativity, diagonalize \(\rho\) in an orthonormal basis. Its eigenvalues \(a_j\) are
nonnegative, and
\(\tr(\rho E_i)=\sum_j a_j h(e_j,E_i e_j)\geq0\).
The normalization follows from \eqref{eq:povm}.
Thus states, effects, measurements, and their finite probability
relations are all definable in the same language.

\subsection{The exact dilation dimension}

For a positive definite space \(W\), a projective measurement is a tuple
of self-adjoint projections \(P_1,\ldots,P_m\) satisfying
\[
P_iP_j=0\ (i\neq j),\qquad \sum_iP_i=I_W.
\]
A dilation of \eqref{eq:povm} consists of an isometry \(J:V\to W\) and
such a tuple with
\[
E_i=J^\dagger P_iJ.
\]
Over \(\mathbb C\) this is the finite-dimensional Naimark construction
\cite{BuschLahtiPellonpaaYlinen2016}. For the explicit minimal dilation
of a qubit POVM, see \citet[Section~3]{PellonpaaDesignolleUola2023}.
The rank formula extends to every real closed fixed field.

\begin{theorem}\label{thm:naimark}
Let \(K=R(u)\) for a real closed field \(R\), and let
\(E_1,\ldots,E_m\) satisfy \eqref{eq:povm} on an \(n\)-dimensional
positive definite space. Set \(r_i=\rank E_i\). The least possible
dimension of a positive definite dilation space is
\begin{equation}\label{eq:naimark-dimension}
D_{\min}=\sum_{i=1}^m r_i.
\end{equation}
Such a dilation exists in every dimension \(d\geq D_{\min}\).
\end{theorem}

\begin{proof}
The positive definite spectral theorem holds over \(K\) by
Theorem~\ref{thm:universal-spectral}. Each \(E_i\) therefore has an
orthonormal eigenbasis and nonnegative real eigenvalues.
Every positive eigenvalue has a square root in \(R\).
Consequently there is a surjective linear map
\[
A_i:V\longrightarrow K^{r_i},
\qquad A_i^\dagger A_i=E_i.
\]
In an eigenbasis for \(E_i\), take the coordinates with positive
eigenvalues and multiply each by the positive square root of its
eigenvalue. When \(r_i=0\), the target is the zero space.
Let
\[
W=\bigoplus_{i=1}^m K^{r_i},
\qquad
Jv=(A_1v,\ldots,A_mv).
\]
Since \(J^\dagger J=\sum_iE_i=I\), the map \(J\) is an isometry.
Let \(P_i\) be the coordinate projection onto the \(i\)-th summand.
Then \(J^\dagger P_iJ=A_i^\dagger A_i=E_i\).

Conversely, let \(J:V\to W\) and \(P_i\) be any dilation.
Set \(A_i=P_iJ\). Positivity of the inner product on \(W\) implies
\[
\ker(A_i^\dagger A_i)=\ker A_i,
\]
because
\(h(v,A_i^\dagger A_iv)=h(A_iv,A_iv)\).
Thus \(\dim(P_iJ(V))=\rank A_i=\rank E_i=r_i\).
The subspaces \(P_iJ(V)\) are mutually orthogonal, so
\(\dim W\geq\sum_i r_i\). This proves minimality.
To obtain a larger dimension, adjoin an orthogonal summand to \(W\),
extend \(J\) by zero into that summand, and assign the new summand to
one projection.
\end{proof}

\begin{example}\label{ex:three-dimensional-dilation}
On \(K^2\), let
\[
E_1=\diag(1,9/25),\qquad E_2=\diag(0,16/25).
\]
Their ranks are \(2\) and \(1\). A minimal dilation on \(K^3\) is
\[
J=\begin{pmatrix}1&0\\0&3/5\\0&4/5\end{pmatrix},
\qquad P_1=\diag(1,1,0),\qquad P_2=\diag(0,0,1).
\]
The equalities \(J^\dagger J=I_2\) and
\(J^\dagger P_iJ=E_i\) hold over every real closed fixed field.
Theorem~\ref{thm:naimark} shows that dimension \(3\) is necessary.
\end{example}

\begin{corollary}[Finite-measurement transfer]\label{cor:measurement-transfer}
Fix the system dimension \(n\), the number \(m\) of outcomes, and the
dilation dimension \(d\). A parameter-free first-order sentence about
rays and their transition probabilities, density operators,
\(m\)-outcome POVMs, outcome probabilities, or dilations into
dimension \(d\) has the same truth value over every \(R(u)\) as over
\(\mathbb C\), with the preceding definitions and the standard positive
definite forms. This truth value is decidable.
\end{corollary}

\begin{proof}
All definitions on the \(n\)-dimensional space are formulas in
\(\LQ\). Since \(d\) is fixed, the additional \(d\)-dimensional
vectors and rectangular maps are encoded by fixed-length scalar
tuples. Their Gram, isometry, and projection conditions are polynomial
equalities together with the definable order conditions.
Theorem~\ref{thm:coordinate-reduction} reduces the resulting sentence
to a parameter-free sentence of \(\RCF\). Completeness and decidability
of \(\RCF\), or equivalently
Corollary~\ref{cor:unframed-hermitian-classification}, give the claim.
\end{proof}

For a particular measurement, the existence of a dilation of dimension
\(d\) is the finite rank condition
\(\sum_i\rank E_i\leq d\). For fixed \(n,m,d\), this condition is a
finite disjunction over the rank tuples satisfying the bound.
\appendix
\section{Explicit formulas and finite calculations}
\label{app:formulas}

\subsection{Linear algebra without named bases}

For a fixed nonnegative integer \(r\), membership in a finitely generated span is
\[
x\in\Span(v_1,\ldots,v_r)
\quad\Longleftrightarrow\quad
\exists a_1\cdots a_r\ x=\sum_i a_iv_i.
\]
Linear dependence is
\[
\exists a_1\cdots a_r
\left(\bigvee_i a_i\neq0\ \wedge\ \sum_i a_iv_i=0\right),
\]
and independence is its negation. An \(n\)-tuple is a basis if and only
if its Gram matrix for the nondegenerate ambient form \(h\) is
invertible. The determinant is a fixed polynomial, so this gives a
quantifier-free basis predicate in \(\LQ\). Alternatively, the dual-basis
criterion of Section~\ref{sec:structures} gives an existential formula.

For an operator \(A\), rank \(r>0\) is specified by independent image
vectors \(Au_1,\ldots,Au_r\) spanning every \(Av\). Rank zero is \(A=0_M\).
Ranks can also be expressed relative to an existentially chosen basis
by the vanishing of all \((r+1)\)-minors and the nonvanishing of some
\(r\)-minor. Nilpotence is \(A^n=0_M\), since the dimension is \(n\).
Self-adjointness is the relation
\[
\forall x\,\forall y\ h(Ax,y)=h(x,Ay).
\]
Equivalently, one can choose a basis \(e_1,\ldots,e_n\)
and check
\[
h(Ae_i,e_j)=h(e_i,Ae_j)\qquad(1\leq i,j\leq n).
\]
Sesquilinearity extends these finitely many identities to all vectors.

Consequently each nilpotent orbit formula in
\eqref{eq:nilpotent-orbit-formula} has an existential version.
For every power form, existentially choose its diagonalizing basis
and use the Gram determinant for the basis condition. On one of
these bases impose the displayed self-adjointness identities.
The condition \(A^n=0_M\) is already atomic.

\subsection{Quotient forms on representatives}

For a definable subspace \(C\leq V\), the cosets
\([v_1],\ldots,[v_s]\) are linearly independent in \(V/C\) when
\[
\forall a_1\cdots a_s\
\left(\sum_i a_iv_i\in C\ \longrightarrow\ \bigwedge_i a_i=0\right).
\]
They span the quotient when
\[
\forall v\ \exists a_1\cdots a_s\
        \left(v-\sum_i a_iv_i\in C\right).
\]
A form on \(V/C\) can therefore be specified and diagonalized by
formulas on representatives, provided it vanishes whenever either
argument lies in \(C\). No additional quotient sort is necessary.
In Section~\ref{sec:pairs}, \(C=\ker N_1=\ker N_2\).
The positive Gram identity
\(h(f_i,N_1f_j)=\delta_{ij}\) already gives independence, and the
known quotient dimension gives spanning.

\subsection{Reproducible finite calculations}

Table~\ref{tab:branch-counts} lists the counts through dimension \(8\).
The program \texttt{anc/verify\_inertia.py} enumerates all \(3{,}131\)
signed partitions in dimensions \(1\) through \(12\). For each signed
block it computes the inertias of its power forms by exact rational
congruence elimination. It sums these inertias over each partition
and checks that \eqref{eq:inertia-inversion} recovers the original
multiplicities. A separate calculation extracts coefficients of
\eqref{eq:branch-generating-function} and agrees with the enumeration
at every signature in this range.

The script also checks the matrix examples, including the pencil at
selected rational parameters and the three-dimensional dilation.
Run it with Python~3 to regenerate the table and the counts by signature.

\begin{table}[htbp]
\centering
\begin{tabular}{rll}
\toprule
\(n\) & \(b_{p,n-p}\) for \(0\leq p\leq\lfloor n/2\rfloor\) & \(\sum_{p=0}^n b_{p,n-p}\) \\
\midrule
1 & 1 & 2 \\
2 & 1, 3 & 5 \\
3 & 1, 4 & 10 \\
4 & 1, 4, 10 & 20 \\
5 & 1, 4, 13 & 36 \\
6 & 1, 4, 14, 27 & 65 \\
7 & 1, 4, 14, 36 & 110 \\
8 & 1, 4, 14, 39, 69 & 185 \\
\bottomrule
\end{tabular}
\caption{Numbers of complete theories with one named self-adjoint nilpotent operator. The omitted signatures follow from \(b_{p,q}=b_{q,p}\).}
\label{tab:branch-counts}
\end{table}

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\end{document}