Inertia and Elementary Theories of Hermitian Nilpotent Operators

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 $-1$.

One operator and two operators

For a self-adjoint nilpotent operator $N$, consider the Hermitian forms

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

with $r_n = r_{n+1} = d_n = d_{n+1} = 0$. Theorem 4.3 proves the formula, and Corollary 4.4 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 5.1). 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 5.3 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 $1 \le s \le\min(p, q)$, 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 6.2). The inertia of

recovers each eigenvalue and its multiplicity. The ranks and inertias of all operator words together can be constant for pairs with distinct spectra (Example 6.5).

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 6.4).

Relation to earlier work

The classification of forms and self-adjoint operators under change of basis is classical [11, 18, 20]. Over $\mathbb{C}$, Mehl [15], Theorem 6.4 gives the signed Jordan blocks used here for the eigenvalue zero. De Seguins Pazzis [8], § 6.1 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 [16 Lemma 1]. Applying this calculation to each power form on $V$ gives (1), 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 Birkhoff and von Neumann [4]. For fixed dimension, see Dunn et al. [9] on quantum logics and Dunn et al. [10], Theorem 2.5 on uniform decidability of the corresponding first-order lattice theories. Xu et al. [24], Theorem 6.2 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, Brunet [5 Theorem 14] proves completeness of a possibilistic theory for each dimension $d \ge3$. In the analytic setting, Argoty Pulido [1] studies Hilbert spaces with a distinguished unbounded closed self-adjoint operator.

The role of the scalar field in quantum theory is discussed by Cassinelli and Lahti [7], Gudder and Piron [12]. Algebraic approaches over other scalar fields include Schumacher and Westmoreland [19], Thas [21]. Thas [22 Sections 3.3 and 4.2] 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 2 and 3 develop the inertia predicates and the coordinate translation used in the classifications of Sections 4–6. Section 7 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 8 proves transfer for finite measurements in the positive definite case and computes the least Naimark dilation dimension. Appendix A gives first-order formulas and finite counts by signature.

Hermitian structures and their first-order language

Throughout, $n \ge1$ is a fixed integer. A field with involution is a field $K$ with an automorphism $\sigma$ satisfying $\sigma^2 = \operatorname{id}$. 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

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.

The scalar fields and inertia

In this situation the order on $R$ is unique and definable. We use the field formulas

The relations $a > b$ and $a \ge b$, for $a,b \in R$, abbreviate $\operatorname{Pos}(a-b)$ and $\operatorname{Pos}(a-b) \lor a = b$. If $z = a + ub$, then

which is positive for $z \ne0$.

For a possibly degenerate Hermitian form $b$, its inertia is the triple $\operatorname{In}(b) = (p(b), q(b), z(b))$ 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 $p(b) - q(b)$.

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

The four sorts

The language $L_{Q,n}$ 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

and a relation $\operatorname{Rep} \subseteq V \times P$. We write $Tv$ for $\operatorname{ev}(T, v)$. No basis is named in $L_{Q,n}$.

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

Add extensionality,

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

These axioms identify $M$ with all of $\operatorname{End}_K(V)$.

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

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 $\operatorname{End}_K(V)$, and the representative relation identifies the point sort with $\mathbb{P}(V)$.

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

In a basis with Gram matrix $H$, it is

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.

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 $a+b\le n$, let $\operatorname{In}_{a,b}(g)$ assert that some basis $v_1,\ldots,v_n$ has Gram matrix

for $g$. The basis condition itself has an existential definition over $\mathcal{T}_n^{\mathrm{str}}$. It asks for vectors $w_1,\ldots,w_n$ satisfying

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 $V$.

Proof. Proposition 2.2 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 $0$, $1$, $-1$.

When a fixed finite-dimensional quotient is used below, its linear independence, basis, and form conditions can likewise be expressed on representatives. Appendix § 9 records the translations.

Coordinate interpretations and semialgebraic reduction

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.

Frames and projective representatives

Fix a symmetric matrix $H \in\mathrm{GL}_n(\mathbb{Z})$. 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

Their solutions are unique. In fact vector coordinates are given by terms,

and the same expression applied to $Te_j$ gives the operator entries.

For projective points, put

Each one-dimensional subspace has a unique representative in $C^{\mathrm{proj}}$, obtained by dividing by the first nonzero coordinate. The formula

therefore specifies a definable bijection between $P$ and this definable subset of $K^n$.

Proof. First replace $V$, $M$, and $P$ by $K^n$, $K^{n^2}$, and $C^{\mathrm{proj}}$, respectively, using (4). Vector operations, matrix multiplication, and evaluation are polynomial. The form is

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

Multiplication and conjugation become

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 (3). 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.

Real closed fields have effective quantifier elimination in the ordered field language [14, 23]. Hence every formula in Theorem 3.1 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 § 5 require no expansion of the fixed field.

Descent and the original language

Proof. An $H$-frame exists in every structure under consideration. The assumed independence makes the existential choice equivalent to any particular choice.

The inertia predicates of Lemma 2.5 are already formulas of $L_{Q,n}$. Proposition 5.5 and Remark 5.6 describe which conclusions about coordinates depend on the named constants.

Inertia of powers and nilpotent unitary orbits

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

Write $(p_j,q_j)$ for its positive and negative indices, and put

The zero index is $n-r_j$. Self-adjointness and nondegeneracy give

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 [15]. We construct its nilpotent blocks over $K=\mathbb{R}(u)$, then compute the contribution of one block to each $B_j^N$. This calculation recovers the signs in any basis.

Signed cyclic decompositions

Let $J_m=J_m(0)$ be the nilpotent Jordan matrix with ones immediately above the diagonal, and let

be the reversal matrix. Thus $F_mJ_m=J_m^{\mathsf{T}}F_m$. 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.

Proof. Since $N^{m-1}\ne0$ and $h$ is nondegenerate, $B_{m-1}^N$ is nonzero. By Proposition 2.2 it has a nonzero diagonal value. Choose $v$ with

Every moment $\beta_j=h(v,N^jv)$ belongs to $\mathbb{R}$, 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 $\mathbb{R}[t]/(t^m)$, put

Its constant term is $1$. Consequently there is a polynomial $P(t)$ with constant term $1$ such that

For example, take the truncated binomial expansion

All its coefficients lie in $\mathbb{R}$. 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 $P(t)^2=\sum_a \gamma_a t^a$ modulo $t^m$ gives

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,

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 $Nx\in W^\perp$.

Proof. Apply Lemma 4.1 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.

Recovery and realization of inertia profiles

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.

These relations are invertible. More precisely, for $1 \le m \le n$,

Consequently the sequence $\left((p_j,q_j)\right)_{j=0}^{n-1}$ determines the signed Jordan multiplicities uniquely.

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

For $j \ge m$ 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 (13)–(14).

Set $t_m=c_m^+ + c_m^-$ and $e_m=c_m^+ - c_m^-$. Taking successive differences in (13) yields

Similarly, the summands in $d_{m+1}$ are exactly those in $d_{m-1}$ except for $e_m$, so

The equalities $c_m^\pm=(t_m\pm e_m)/2$ give (15). The specified zero boundary values cover $m=n$ and $m=n-1$ as well. ∎

First-order definitions of the unitary orbits

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

Proof. If $N' = gNg^{-1}$ with $g \in\mathrm{U}(h)$, then

so every inertia is preserved. Conversely, equality of the inertias and Theorem 4.3 give equality of all signed block multiplicities. Match the signed block bases supplied by Theorem 4.2. The resulting linear map preserves $h$ and intertwines $N$ with $N'$, proving conjugacy.

For the definability assertion, apply Lemma 2.5 to $B_j^T$. With $\Delta^{a,b}=\operatorname{diag}(I_a,-I_b,0_{n-a-b})$ the formula is

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

Here $\operatorname{Herm}(T)$ abbreviates $\forall x\ \forall y\ h(Tx,y)=h(x,Ty)$. The first part of the theorem shows that (20) 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 $n$. $\square$

The disjunction of $\Phi_{\mathcal{I}}(T)\land\Phi_{\mathcal{I}}(S)$ over the profiles satisfying (17) thus defines unitary conjugacy on the nilpotent self-adjoint operators.

The largest nilpotency index

Elementary classification and definable sets

We use the original multisorted language $L_{Q,n}$, without a named basis or imaginary unit, and the theories $T_n^{\mathrm{inv}}$ and $T_n^{p,q}$ from Definition 2.3. In the language $L_{Q,n}(N)$, where $N$ is a constant of the operator sort, put

Here the adjoint notation abbreviates its defining relation. Adding the signature sentence $\operatorname{Sig}_{p,q}$ gives $T_{n,\mathrm{nil}}^{p,q}$, where $p+q=n$.

Defining the signed Jordan branches

Write

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 § 4.1. Thus $H_\gamma$ is symmetric, integral, and has determinant $\pm1$. Let $(p_\gamma,q_\gamma)$ be its signature. Define

The invertible Gram matrix makes the $n$ vectors in $\bar e$ a basis. Set

Equivalently, one can specify a branch by the power forms $B_j^N(x,y)=h(x,N^jy)$, using the inertia predicates from Lemma 2.5. If $(p_j^\gamma,q_j^\gamma)$ are the inertias of $H_\gamma J_\gamma^j$, Theorems 4.2 and 4.3 give

Thus $\chi_\gamma$ can be read either as a condition on a single Jordan basis or as a list of basis-independent inertia conditions.

Proof. The signed decomposition in Theorem 4.2 provides a $\gamma$ for each model. The inversion formula (15) 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 3.1 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 $L_{Q,n}(N)$ has on this branch a parameter-free ordered-field translation. Completeness of RCF therefore proves completeness of the branch. Together with (22), 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

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

Indeed, positive elements of a real closed field are its nonzero squares. This embedding is elementary by model completeness of 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 3.1 thus gives, for every formula $\varphi(\bar x)$ and every tuple $\bar a$ from $\mathcal{A}$,

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 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).

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.

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.

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.

Semialgebraic definable sets and the fixed field

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 3.1 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 [14, 23]. 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.

In the ordered-field coordinates, positivity is atomic. In $L_{Q,n}$ it requires a quantifier, even if the frame and imaginary unit are named.

Proof. Consider the definable subset

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 $T^{p,q}_{n}$.

Remark 5.6. 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.

Pairs of square-zero operators and Hermitian pencils

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 $1\le s\le\min(p,q)$.

Definition 6.1. A positive pair of rank $s$ consists of two named operators $N_1,N_2$ in a model of $T^{p,q}_{n}$ such that

and

The resulting first-order theory is denoted $T^{p,q;s}_{\mathrm{pair}}$.

Set $W = \operatorname{im} N_{1} = \operatorname{im} N_{2}$. Self-adjointness gives $\ker N_{i} = (\operatorname{im} N_{i})^{\perp} = W^{\perp}$, and $N_{i}^{2} = 0$ gives $W \subseteq W^{\perp}$. Thus

On the $s$-dimensional quotient $E = V/W^{\perp}$, define

Changing either representative by an element of $W^{\perp}$ does not change this expression. The forms are Hermitian and nondegenerate. Condition (24) says exactly that both are positive definite.

A simultaneous normal form

Proof. Let $A \in\operatorname{End}_{K}(E)$ be defined by

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 $e_{i} = N_{1}f_{i}$. Then

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

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 (26).

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 (26) and produces a simultaneous isometry.

The simultaneous reduction of two forms with one positive definite form is the usual definite Hermitian pencil construction [11, 13]. Theorem 6.2 realizes that construction inside the square-zero operator geometry, including the common radical and the ambient orthogonal complement.

The complete theories of positive pairs

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

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 (28) and the interpreted fixed-field order. These are the two claimed Turing reductions.

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.

Witt index and orthogonal spectral decompositions

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.

Proof. If $N=N^\dagger$ and $N^2=0$, then

so $\operatorname{im}N$ is totally isotropic.

Conversely, let $W=\operatorname{span}(e_1,\ldots,e_s)$ be totally isotropic and put

This map is linear, and

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 $\operatorname{im}N=W$. For $s=0$, take $N=0$.

For fixed $n$, this theorem gives first-order sentences

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 § 2.

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

For fixed $n$, this is a first-order predicate $\operatorname{COEB}(T)$.

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.

Finite measurements over real closed fixed fields

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

The expression is independent of the representatives. Completing a square,

shows $0 \le\mathrm{P} \le1$ in $R$. Its graph is first-order, after clearing the nonzero denominator. The corresponding rank-one orthogonal projection is

States and effects

An operator $A$ is positive semidefinite, written $A \succeq0$, if

A density operator satisfies $\rho\succeq0$ and $\operatorname{tr}\rho=1$. 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

These are the finite-dimensional positive-operator-valued measurements (POVMs). For a density operator $\rho$, the outcome probabilities are

They lie in $R_{\ge0}$ and sum to $1$. To see nonnegativity, diagonalize $\rho$ in an orthonormal basis. Its eigenvalues $a_j$ are nonnegative, and $\operatorname{tr}(\rho E_i)=\sum_j a_jh(e_j,E_i e_j)\ge0$. The normalization follows from (31). Thus states, effects, measurements, and their finite probability relations are all definable in the same language.

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

A dilation of (31) consists of an isometry $J:V\to W$ and such a tuple with

Over $\mathbb{C}$ this is the finite-dimensional Naimark construction [6]. For the explicit minimal dilation of a qubit POVM, see Pellonpää et al. [17, Section 3]. The rank formula extends to every real closed fixed field.

Proof. The positive definite spectral theorem holds over $K$ by Theorem 7.3. 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

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

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

because $h(v,A_i^\dagger A_iv)=h(A_iv,A_iv)$. Thus $\dim(P_iJ(V))=\operatorname{rank}A_i=\operatorname{rank}E_i=r_i$. The subspaces $P_iJ(V)$ are mutually orthogonal, so $\dim W\ge\sum_i r_i$. 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.

Proof. All definitions on the $n$-dimensional space are formulas in $L_{Q,n}$. 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 3.1 reduces the resulting sentence to a parameter-free sentence of RCF. Completeness and decidability of RCF, or equivalently Corollary 5.2, give the claim.

For a particular measurement, the existence of a dilation of dimension $d$ is the finite rank condition $\sum_i \operatorname{rank} E_i \le d$. For fixed $n$, $m$, $d$, this condition is a finite disjunction over the rank tuples satisfying the bound.

Explicit formulas and finite calculations

Linear algebra without named bases

For a fixed nonnegative integer $r$, membership in a finitely generated span is

Linear dependence is

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 $L_{Q,n}$. Alternatively, the dual-basis criterion of § 2 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

Equivalently, one can choose a basis $e_1,\ldots,e_n$ and check

Sesquilinearity extends these finitely many identities to all vectors. Consequently each nilpotent orbit formula in (20) 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.

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

They span the quotient when

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 § 6, $C=\ker N_1=\ker N_2$. The positive Gram identity $h(f_i,N_1f_j)=\delta_{ij}$ already gives independence, and the known quotient dimension gives spanning.

Reproducible finite calculations

Table 1 lists the counts through dimension 8. The program 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 (15) recovers the original multiplicities. A separate calculation extracts coefficients of (23) 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.

$n$	$b_{p,n-p}$ for $0 \leq p \leq\lfloor n/2\rfloor$	$\sum_{p=0}^{\lfloor n/2\rfloor} b_{p,n-p}$
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