% Virasoro constraints under projectivization - complete, self-contained.
% Rewritten for readability (Section 1 is a plain-language roadmap).
% All lemmas and proofs from the original are preserved.

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\title{Virasoro Constraints Under Projectivization}
\author{}
\date{}
\begin{document}
\maketitle

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

\subsection{What is being proved, informally}

Let $Y$ be a smooth complex projective variety. Gromov--Witten
(GW) theory ``counts'' maps from compact curves into $Y$; each
such map contributes a number that also remembers two pieces of
data:
\begin{itemize}
\item a homology class $d\in H_2(Y;\Z)$ telling us which curve
class of $Y$ we are counting;
\item at each of the $n$ marked points $x_i$ on the domain
curve, a cohomology class $b_i\in H^*(Y;\C)$ together with a
power $\psi_i^{k_i}$ of the first Chern class of the cotangent
line at $x_i$.
\end{itemize}
The resulting \emph{descendant integral} is
\[
 \int_{[\overline{\mathcal M}_{g,n}(Y,d)]^{\mathrm{vir}}}
 \psi_1^{k_1}\ev_1^* b_1\cdots
 \psi_n^{k_n}\ev_n^* b_n.
\]
Stacking \emph{every} such number (all $g$, $d$, $n$, all
$k_i$, all $b_i$) into a single formal generating series gives
the \emph{total descendant potential} $Z_Y(t)$, defined
precisely in Section~\ref{sec:conv}.

The \emph{Virasoro constraints} are a sequence of differential
equations
$L_k^Y Z_Y=0$, $k\geq -1$, in the sources $t$.
They were proposed by Eguchi--Hori--Xiong and refined by
Getzler, and are known for target curves, toric targets, and
targets with semisimple quantum cohomology, but remain open in
general. This paper proves that they pass from a base to the
projectivization of any rank $\geq 2$ vector bundle over it. The
transfer needs no semisimplicity of quantum cohomology and no
positivity or splitness of the bundle.

The precise statement is:

\begin{theorem}\label{thm:main}
Let $B$ be a smooth connected projective variety over $\C$,
let $E$ be an algebraic vector bundle of rank $r\geq 2$ on
$B$, and set $X=\mathbb P_B(E)$. If the ordinary descendant
potential of $B$ is annihilated by every Virasoro mode,
$L_k^B Z_B=0$ for all $k\geq -1$, then the same holds for $X$:
$L_k^X Z_X=0$ for all $k\geq -1$.
\end{theorem}

Here $Z_Y$ is the \emph{ordinary} descendant potential of $Y$
(no ancestor classes, no twisting), on the full cohomology
$H^*(Y;\C)$, with each actual integral curve class kept
separate. The conclusion includes every genus, every individual
integral curve class, every finite list of descendants, and
primitive and odd insertions. Applying the theorem repeatedly
covers towers $X\to X'\to\cdots\to B$ of projective bundles
over any base satisfying the hypothesis.

\subsection{Why this is not obvious}

If $E$ splits as $L_1\oplus \cdots\oplus L_r$, mirror-symmetry
arguments prove the transfer (Coates--Givental--Tseng), and if
$B$ has semisimple quantum cohomology, semisimplicity transfers
to $X$ (Iritani--Koto) and Givental--Teleman reconstruction then
gives the constraints on $X$. Neither assumption is available in
general. Non-split bundles and non-semisimple bases require an
argument that does not go through mirror symmetry.

Our proof uses a model geometry that sees both $B$ and $X$ as
fixed loci, plus a $\C^*$-equivariant shift operator that turns
the ``change of fixed component'' into ``change of spectral
block.'' The shift and its spectrum do not depend on
semisimplicity. A separate step (Quantum Riemann--Roch) carries
the constraints from the local neighborhood of each fixed
component to its ordinary theory. The sum over the fiber's
spectral cluster assembles the constraints on $X$.

\subsection{The master space and the five ideas}

Twist $E$ by a sufficiently negative line bundle on $B$. This
does not change $X=\mathbb P_B(E)$, but makes $E^*$ globally
generated, hence makes tautological degrees nonnegative on
effective curves. Set
\[
 W=\mathbb P_B\bigl(E\oplus\mathcal O_B\bigr).
\]
Let $T=\C^*$ act with weight $0$ on $E$ and weight $1$ on
$\mathcal O_B$, and let $\lambda\in H_T^2(\text{pt})$ be the
equivariant parameter. The fixed scheme is a disjoint union
$W^T=B\sqcup X$: the section $[0:1]$ is $B$, and the section
$[1:0]$ is $X$.

The proof has five steps:

\begin{enumerate}
\item \textbf{Localize the master ancestors.} The equivariant
descendant theory of $W$ factorizes, via the stable-graph sum of
virtual localization, into the two fixed-component theories,
each twisted by the inverse Euler class of its normal bundle,
plus an upper-triangular symplectic transformation $R(z)$: see
Proposition~\ref{prop:loc-factorization} below. The $R$ and its
inverse are explicitly quantizable.

\item \textbf{Identify the Iritani shift and its spectrum.}
Iritani's equivariant shift operator $T_a$ translates
$\lambda\mapsto\lambda+z$. It has a closed formula
(T_a=S_WT_dS_W^{-1}) and, on the fiber part of the moderate
cohomology, its leading term is multiplication by $p+\lambda$
in the quantum algebra of the fiber. Hence the distinct
eigenvalue branches are $\lambda+h$, with
$h^r(h+\lambda)=y$ and $y$ the fiber class. There are
$r+1$ branches; near $y=0$, one tends to $0$ (the \emph{$B$
branch}), and the other $r$ tend to $\lambda$ (the \emph{$X$
cluster}).

\item \textbf{Isolate spectral blocks.} Contour integrals against
the resolvent of $T_a$ produce projectors $P_i$ and logarithms
$\mathscr L_i$ on each branch. Because the graded operator
$G_a$ (which has $\lambda\partial_\lambda$ in it) commutes with
$T_a$, the formula
$D_i=P_iG_a-(\lambda/z)\mathscr L_i$ has no
$\lambda$-differentiation. The induced loop operators on blocks
are therefore quantizable over the coefficient ring.

\item \textbf{Continue around the branch cover.} The cover
$h\mapsto h^r(h+\lambda)$ is connected away from its critical
values, so monodromy in $y$ transports every equation from the
$B$ branch to every other branch. Ancestor coefficients are
polynomial in $y$, and their cotangent powers are bounded
independently of fiber degree; consequently the interchange of
``formal identity'' and ``identity of germs'' is legal
coefficientwise. Adding the $r$ branches of the $X$ cluster
gives the constraints on $X$ after going back to $y=0$.

\item \textbf{Fix the scalar normalization.} Quantized transports
are only defined projectively. Two commutators,
$[L_{-1}^Y,L_1^Y]=-2L_0^Y$ and $[L_0^Y,L_k^Y]=-kL_k^Y$, remove
the insertion-independent scalars and give exactly the
prescribed constant $C_Y$ in $L_0^Y$.
\end{enumerate}

Two ingredients of the argument are worth flagging, since they
are reusable:
(1) subtracting a block logarithm from a differential operator
in $\lambda$ converts a derivative into a loop operator over the
coefficient ring; and
(2) ancestor bounds make every coefficient test finite, so
analytic continuation between branches is legitimate
coefficientwise.
Neither ingredient is special to projective bundles.

\subsection{How to read this paper}

Section~\ref{sec:conv} fixes conventions once and for all:
descendant potentials, loop space, ordinary modes, normal
ordering, projective constraints, and coefficientwise finiteness.
Section~\ref{sec:geo} constructs the master space, its fixed
components, and the pairings between them. Section~\ref{sec:loc}
proves the localization factorization of ancestors on $W$.
Section~\ref{sec:shift} defines the calibrated grading and the
Iritani shift, computes the fiber spectrum, and records the
commutation $[G_a,T_a]=0$. Section~\ref{sec:spec} constructs the
block projectors and logarithms and their modes.
Section~\ref{sec:fixed} identifies, via Quantum Riemann--Roch,
the block operators on each fixed component with the ordinary
Virasoro operators. Section~\ref{sec:cont} handles monodromy,
scalar normalization, and finishes the proof of
Theorem~\ref{thm:main}.

Unproved but citable facts are limited to: virtual localization
(Graber--Pandharipande), Iritani's equivariant shift theorem,
Quantum Riemann--Roch (Coates--Givental), and Givental's upper
triangular quantization formula. All other statements are proved
in place, with the argument written out.

% ==================================================================
\section{Conventions: descendants, quantization, Virasoro}\label{sec:conv}
% ==================================================================

Throughout the paper, $Y$ is a smooth connected projective
variety over $\C$ of dimension $d$. The two examples to keep in
mind are $Y=B$ and $Y=X=\mathbb P_B(E)$; every definition in
this section applies verbatim to both.

\subsection{State space and grading}

The state space is the full super-vector space
$H_Y=H^*(Y;\C)$ with parity given by cohomological degree
modulo $2$ and Poincar\'e pairing
$\eta_Y(a,b)=(a,b)_Y=\int_Y a\cup b$. Tensor products,
permutations, contractions and derivatives follow the Koszul
rule; the coevaluation tensor is the categorical inverse of
$\eta_Y$, without an extra parity sign.

Two even endomorphisms of $H_Y$ organize the Virasoro
construction:
\begin{equation}
 \mu_Y\big|_{H^{a,b}(Y)}=(a-d/2)\id,
 \qquad
 R_Y=c_1(TY)\cup(-).
 \label{eq:grading}
\end{equation}
Here $a$ is the first Hodge index, so $\mu_Y$ uses the first
Hodge grading (which is the correct choice for targets with odd
cohomology; see Getzler). Poincar\'e duality gives
$\mu_Y^*=-\mu_Y$ and $R_Y^*=R_Y$, and the Hodge type of $c_1$
gives $[\mu_Y,R_Y]=R_Y$.

\subsection{Descendant potentials and Novikov completion}

Fix a homogeneous Hodge basis $(\phi_a)$ with $\phi_0=1_Y$. Let
$t_j^a$ be formal supercommuting variables with the parity of
$\phi_a$, $j\geq 0$, and put
$t_j=\sum_a t_j^a\phi_a$, $t(z)=\sum_{j\geq 0}t_jz^j$. Define
the connected ordinary unreduced descendant potentials and
total potential by
\begin{equation}
 F_g^Y(t)
 =\sum_{\substack{\beta\ \text{effective}\\m\geq 0}}
     \frac{Q^\beta}{m!}
     \int_{[\overline{\mathcal M}_{g,m}(Y,\beta)]^{\mathrm{vir}}}
     \prod_{i=1}^m\Bigl(\sum_{j\geq 0}\psi_i^j
     \ev_i^* t_j\Bigr),
 \qquad
 Z_Y(t)=\exp\sum_{g\geq 0}\hbar^{g-1}F_g^Y(t).
 \label{eq:potential}
\end{equation}
Only stable maps contribute. In particular, $\psi_i$ is the map
cotangent class at the $i$-th mark, not an ancestor class. The
exponential records disconnected domains and the empty domain.

Novikov variables remember the actual effective integral class
$\beta\in H_2(Y;\Z)$, not just its numerical degree:
$Q^{\beta_1}Q^{\beta_2}=Q^{\beta_1+\beta_2}$ and $\beta=0$
corresponds to $Q^0=1$. Fix any ample integral class on $Y$ to
give the Novikov completion: only finitely many effective
$\beta$ lie below any fixed ample degree. All identities below
are read coefficientwise in this completion, at finite order in
the descendants, and coefficientwise in $\hbar$ (which is a
formal Laurent variable).

\subsection{Loop space and ordinary Virasoro modes}

Set
\[
 \cH_Y=H_Y((z^{-1})),\quad
 \Omega_Y(f,g)=\Res_{z=0}\bigl(f(-z),g(z)\bigr)_Y\,dz,
\]
with polarization
$\cH_Y=H_Y[z]\oplus z^{-1}H_Y[[z^{-1}]]$.
The infinitesimal symplectic operators are
\begin{equation}
 \ell_{-1,Y}=z^{-1},\quad
 \ell_{0,Y}=z\partial_z+\tfrac12+\mu_Y+\frac{R_Y}{z},\quad
 \ell_{k,Y}=\ell_{0,Y}(z\ell_{0,Y})^k,\quad k\geq 1.
 \label{eq:modes}
\end{equation}
The symplectic identity is exactly the adjoint triple
$\mu_Y^*=-\mu_Y$, $R_Y^*=R_Y$, $[\mu_Y,R_Y]=R_Y$ combined with
the sign of $z$ in the residue pairing; we check this in the
proof of Lemma~\ref{lem:commutators} below, where we use it.

\subsection{Quantization and Virasoro operators}

Write $q_j^a$ for the positive loop coordinates (dual to
$t_j^a$ in the standard positive part) and $p_{j,a}$ for the
dual negative coordinates. Quantize a quadratic Hamiltonian
$\mathcal Q_A(f)=\frac12\Omega_Y(f,Af)$ by the rule: a $qq$
monomial becomes $q^\top q/(2\hbar)$, a $pq$ monomial becomes
the first-order operator with left multiplication before left
derivative, and a $pp$ monomial becomes $\hbar$ times the
second derivative; all reorderings have their graded signs.
Then perform the dilaton translation
\begin{equation}
 q(z)=t(z)-z1_Y.
 \label{eq:dilaton}
\end{equation}
Denote the resulting operator by $\op_{\eta_Y}(A)$.

With these conventions:
\begin{align}
 L_{-1}^Y
 &=\op_{\eta_Y}(\ell_{-1,Y})
  =-\frac{\partial}{\partial t_0^0}
  +\sum_{j\geq 0,a}t_{j+1}^a\frac{\partial}{\partial t_j^a}
  +\frac{(t_0,t_0)_Y}{2\hbar},
 \label{eq:string}\\
 L_k^Y&=\op_{\eta_Y}(\ell_{k,Y})\quad(k\neq 0),
 \qquad
 L_0^Y=\op_{\eta_Y}(\ell_{0,Y})+C_Y,
 \label{eq:normalized-modes}\\
 C_Y&=\frac{\chi(Y)}{16}-\frac14\str(\mu_Y^2).
 \nonumber
\end{align}
The supertrace counts even minus odd cohomological parity.
Positive modes have no scalar correction.

Write $\V(Y)$ for the conjunction of identities
\begin{equation}
                     L_k^Y Z_Y=0,\qquad k\geq -1.
 \label{eq:virasoro}
\end{equation}
These identities are read coefficientwise: separately at each
genus, at each actual integral class, at each number of marks,
and at each monomial in the descendants. The main theorem is
$\V(B)\Rightarrow\V(X)$.

\subsection{Projective satisfaction and covariance under frame change}

Quantization of an infinitesimal symplectic mode is defined only
up to an additive scalar. Consequently a change of frame can
change the mode equation by a constant. For transfer purposes
we track the weaker projective version, then remove the scalar
ambiguity at the end.

\begin{definition}[Projective satisfaction]\label{def:projective}
An even infinitesimal symplectic $A$ is \emph{satisfied
projectively} by $Z$ if $Z^{-1}\op_\eta(A)Z$ is independent of
the descendants (it may still depend on the Novikov variables,
$\lambda$, and $\hbar$).
\end{definition}

\begin{lemma}[Projective covariance]\label{lem:covariance}
Let $M$ be a symplectic map between paired loop spaces
$(\cH_Y,\eta_Y)$ and $(\cH_{Y'},\eta_{Y'})$, with both $M$ and
$M^{-1}$ admitting quantized actions $U_M,U_{M^{-1}}$ in the
coefficientwise completion. If $A$ and $MAM^{-1}$ are
infinitesimal symplectic and quantizable, then
\begin{equation}
 U_M\op_{\eta_Y}(A)U_M^{-1}
  =\op_{\eta_{Y'}}(MAM^{-1})+\text{scalar}.
 \label{eq:covariance}
\end{equation}
In particular, $A$ is satisfied projectively by $Z$
iff $MAM^{-1}$ is satisfied projectively by $U_MZ$; multiplying
either potential by an invertible scalar does not affect the
projective condition.
\end{lemma}

\begin{proof}
The commutator of two normally ordered quadratic operators is
the quantization of their Hessian/Hamiltonian bracket plus a
scalar double contraction, which in super Darboux coordinates
is a supertrace. Exponentiating this identity gives
\eqref{eq:covariance}; equivalently this is Givental's
projective-quantization formalism. Neither $U_M$ nor $MAM^{-1}$
differentiates $\hbar$ or the parameters, so the projective
condition is preserved.
\end{proof}

\subsection{Coefficientwise finiteness}

Some of the mode matrices defined below have infinitely many
positive powers of $z$, or finite-order $z\partial_z$
derivatives. The following lemma collects the formal facts we
need.

\begin{lemma}[Coefficientwise finiteness]\label{lem:finiteness}
Suppose a potential has finitely many descendant coefficients
at each fixed curve class, $\hbar$-power, and total descendant
degree, and has $\hbar$-powers bounded below at each fixed curve
class and total degree. Let $A$ be an even infinitesimal
symplectic mode with a finite $z$-lower-bound and finite order in
$z\partial_z$ at every filtration step. Then $\op_\eta(A)$ acts
coefficientwise on the potential. The same holds with additional
formal filtration variables; and quantized actions $U_M$ of
transformations equal to the identity modulo positive Novikov
filtration are defined by their formal exponential and inverse
series.
\end{lemma}

\begin{proof}
At fixed filtration, the $pp$ part differentiates only finitely
many coefficients. In the $pq$ part a bounded differentiated
index and a lower bound on loop powers leaves only finitely many
multiplied indices. In the $qq$ part the order in $z\partial_z$
is finite. Euler differentiation $z\partial_z$ multiplies mode
coefficients by polynomials in their indices and preserves these
bounds. At fixed positive filtration, only finitely many terms of
an exponential can occur.
\end{proof}

Ordinary descendant potentials satisfy the required support:
for fixed genus, class and number of marks, high enough powers
of $\psi_i$ vanish on the finite-dimensional stable-map space.
Ancestor potentials satisfy a stronger bound from the
stable-curve moduli space (Lemma~\ref{lem:ancestor-bound}
below).

\paragraph{Categorical inverse and signs.} All splitting and
cotangent-class identities used later are tensor identities
built from the categorical inverse of $\eta_Y$. Their standard
marked-point proofs therefore apply to full cohomology with
these signs. Quantum Riemann--Roch and the two-mark shift
identities likewise use arbitrary evaluation classes and this
diagonal.

% ==================================================================
\section{The master space and its fixed theories}\label{sec:geo}
% ==================================================================

We put $B$ and $X=\mathbb P_B(E)$ into one smooth projective
variety $W$ with a torus action, so that both are fixed loci.
Virtual localization will then express the theory of $W$ in
terms of the theories of $B$ and $X$. This section sets up the
geometry, curve labels, pairings, and the ancestor formalism.

\subsection{Geometry and integral curve labels}

Tensoring $E$ by a line bundle on $B$ does not change
$X=\mathbb P_B(E)$. Choose such a twist so that $E^*$ is
globally generated, and (abusing notation) continue to write
$E$ for the twisted bundle. Set
\[
 X=\mathbb P_B(E),\qquad
 W=\mathbb P_B\bigl(E\oplus\mathcal O_B\bigr).
\]
Let $T=\C^*$ act with weight zero on $E$ and weight one on
$\mathcal O_B$. Write
$\lambda\in H_T^2(\mathrm{pt})$ for the weight-one
representation's class. The fixed scheme is
$W^T=X\sqcup B$, where the second component is the section
corresponding to the summand $\mathcal O_B$.

Write $\pi$ for the projective-bundle projection of either $X$
or $W$ to $B$, and put $p=c_1^T(\mathcal O_W(1))$. The normal
bundles, their ranks, weights, and the integers $W_F,j_F$ that
will appear in the shift formula below are:
\begin{equation}
\begin{array}{c|c|c|c|c|c}
 F&N_F&n_F=\rk N_F&w_F&W_F=n_Fw_F&j_F\\ \hline
 X&\mathcal O_X(1)&1&+1&+1&0\\
 B&E&r&-1&-r&1
\end{array}
 \label{eq:normal-data}
\end{equation}
The restrictions of the equivariant divisor are
\[
              p|_X=c_1(\mathcal O_X(1)),\qquad p|_B=-\lambda.
\]
Here $w_F$ is the common $\C^*$-weight on the normal fibers.
Both descriptions come from the standard relative tangent
sequence: at a line $L\subset V$ one has
$T_{W/B}\cong\Hom(L,V/L)$.

\begin{lemma}\label{lem:curve-labels}
For $P=X$ or $P=W$, the map
\begin{equation}
 H_2(P;\Z)\to H_2(B;\Z)\oplus\Z,\quad
 d\mapsto\Bigl(\pi_*d,\int_d c_1(\mathcal O_P(1))\Bigr)
 \label{eq:integral-splitting}
\end{equation}
is an isomorphism, including torsion. An effective class has
effective base pushforward and nonnegative tautological degree.
\end{lemma}

\begin{proof}
The structure group of a projective bundle acts trivially on
the integral homology of the fiber, so the Serre spectral
sequence in total degree two is
$H_2(B;\Z)\oplus H_2(\mathbb P^{s-1};\Z)$ with $s\geq 2$ the
rank. The fiber generator survives integrally because
$c_1(\mathcal O_P(1))$ pairs to one with the fiber line: its
image in total homology cannot be zero or a nontrivial multiple.
Thus we have an exact sequence
\[
 0\to\Z[\text{fiber line}]\to H_2(P;\Z)
   \xrightarrow{\pi_*}H_2(B;\Z)\to 0,
\]
and pairing with $c_1(\mathcal O_P(1))$ provides a retraction on
the first term, giving \eqref{eq:integral-splitting}. Finally,
$E^*$ and $E^*\oplus\mathcal O_B$ are globally generated, so the
tautological line bundle is globally generated and has
nonnegative degree on effective curves; proper pushforward
preserves effectivity.
\end{proof}

Henceforth we write $Q^\beta y^l$ for the actual class
$(\beta,l)$. Under $X\hookrightarrow W$ the pair $(\beta,l)$ is
unchanged; under $B\hookrightarrow W$ a class $\beta$ becomes
$(\beta,0)$.

\paragraph{Completion.} Fix an ample integral class $H$ on $B$;
for $M$ large enough, $c_1(\mathcal O_P(1))+M\pi^*H$ is ample on
both projective bundles. We complete in the degree
$l+M\int_\beta H$. Only finitely many effective labels occur
below each fixed degree. Unless otherwise noted, ``positive
Novikov filtration'' means positive degree in this completion
(so it includes positive fiber degree even for zero base class).

\subsection{Equivariant pairings and inverse-Euler twists}

The equivariant projective-bundle formula makes $H_T^*(W)$ free
over $\C[\lambda]$ with basis
$\pi^*\phi_a\cdot p^j$, $0\leq j\leq r$, for a homogeneous
Hodge basis $(\phi_a)$ of $H_B$. Equivariant integration pairs
this basis perfectly over $\C[\lambda]$: fiberwise integration
gives an antitriangular matrix whose antidiagonal blocks are
the ordinary Poincar\'e pairing of $B$.

After extending scalars to $\C(\lambda)$, restriction to the
fixed locus is an isomorphism of paired spaces
\begin{equation}
 \bigl(H_T^*(W),\eta_W\bigr)\otimes_{\C[\lambda]}\C(\lambda)
   \cong
 \bigoplus_{F=B,X}
   \bigl(H_F\otimes\C(\lambda),\eta_F^t\bigr),
\qquad
 \eta_F^t(a,b)=\int_F\frac{a\cup b}{e_T(N_F)}.
 \label{eq:fixed-pairing}
\end{equation}
The unit restricts to the sum of the two fixed units. The
Euler classes are invertible over $\C(\lambda)$ because both
normal weights are nonzero.

The corresponding \emph{inverse-Euler-twisted} Gromov--Witten
theory on $F$ inserts
\begin{equation}
 e_T\!\left(R^\bullet\rho_*f^*N_F\right)^{-1}
 \label{eq:Euler-twist}
\end{equation}
in each stable-map integral, extended multiplicatively to the
indicated $K$-theory class. Here $\rho$ and $f$ are the
universal curve and universal map. We use subscripts
$F,\mathrm{tw}$ for its potentials and fundamental solutions.
Its degree-zero three-point pairing is exactly $\eta_F^t$.

\paragraph{Hodge bidegrees.} Give $p$ and $\lambda$ Hodge
bidegree $(1,1)$. In a homogeneous polynomial basis, $\mu$ acts
on basis vectors by the first Hodge degree minus $\dim_\C/2$,
and differentiation in $\lambda$ records the degree of
equivariant coefficients. Virtual dimension identities can be
expressed in this first degree: algebraicity of evaluation
maps and the virtual class forces the sum of Hodge-degree
differences of insertions to be zero in a nonzero invariant.
This remains true equivariantly (check on finite-dimensional
approximations or after fixed-locus integration). This is a
separate grading from the cohomological degree that later
bounds the fiber degrees.

\subsection{Ancestors and the fundamental solution}\label{sec:ancestors}

We now fix notation for ancestors, applicable to an ordinary
theory, an equivariant theory, or one of the twisted fixed
theories. Denote its pairing by $\eta$, its Novikov monomial
for class $d$ by $\mathsf q^d$, its stable-map integrals by
brackets. Let $u$ be a formal even primary background. For
distinguished insertions $a_1,\ldots,a_m$ set
\[
 \langle a_1,\ldots,a_m\rangle_{g,u}
 =\sum_{\substack{d\ \mathrm{eff.}\\n\geq 0}}
    \frac{\mathsf q^d}{n!}
    \bigl\langle a_1,\ldots,a_m,
      \underbrace{u,\ldots,u}_{n}\bigr\rangle_{g,m+n,d}.
\]
Here $d$ runs over effective classes and only stable-map terms
are included. Descendants in these brackets use map cotangent
classes.

For $2g-2+m>0$ there is a stabilization map
\[
 \operatorname{st}_{g,m}:
 \overline{\mathcal M}_{g,m+n}(Y,d)\to
 \overline{\mathcal M}_{g,m}
\]
forgetting the map and the $n$ background marks. Set
$\bar\psi_i=\operatorname{st}_{g,m}^*\psi_i$, $1\leq i\leq m$.
The ancestor potential at background $u$ is
\begin{equation}
 \overline F_g(u;t)
 =\sum_{\substack{d\ \mathrm{eff.},\ m,n\geq 0\\2g-2+m>0}}
    \frac{\mathsf q^d}{m!n!}\Bigl\langle
     \prod_{i=1}^m\hspace{-3pt}\Bigl(\sum_{j\geq 0}
       \bar\psi_i^j\ev_i^* t_j\Bigr)
     \prod_{a=m+1}^{m+n}\ev_a^* u\Bigr\rangle_{g,m+n,d},
\quad
\mathcal A(u;t)=\exp\sum_{g\geq 0}\hbar^{g-1}\overline F_g(u;t).
 \label{eq:ancestor-potential}
\end{equation}
The bracket denotes integration of the indicated product,
with the twist when appropriate. Terms with curve-unstable
distinguished data are omitted even when the stable-map space
itself is nonempty; in particular the genus-one term with no
distinguished marks is omitted. Ancestors come from
$\overline{\mathcal M}_{g,m}$, so a total power above
$3g-3+m$ vanishes.

\begin{definition}[Fundamental solution]\label{def:fundamental}
The fundamental solution $S(u,z)=\id+O(z^{-1})$ is defined by
\begin{equation}
 \eta(b,S(u,z)a)=\eta(b,a)
   +\Bigl\langle b,\frac{a}{z-\psi}\Bigr\rangle_{0,u},
\qquad
 \frac{1}{z-\psi}=\sum_{j\geq 0}\psi^j z^{-j-1}.
 \label{eq:S-definition}
\end{equation}
The pairing term supplies the unstable two-point
contribution.
\end{definition}

The genus-zero splitting and cotangent-class identities give
\begin{equation}
 S(u,-z)^*S(u,z)=\id,\qquad
 z\partial_v S(u,z)=(v\star_u)S(u,z),
 \label{eq:S-identities}
\end{equation}
where $\star_u$ is the quantum product and $\partial_v$
differentiates the primary background. Recall that the
genus-zero cone $\mathcal L$ is the formal Lagrangian graph of
$dF_0$ in Darboux coordinates $(q,p)$ with $q=t-z1$. Its point
with descendant input $\epsilon$ has positive coordinate
$\epsilon-z1$, and the tangent space there is the graph of the
Hessian of $F_0$. The string, dilaton and topological recursion
identities make this tangent space tangent along $zT\subset
\mathcal L$; at primary coordinate $u$ the tangent space is
$S(u,z)^{-1}\cH_+$ (Givental). The ancestor--descendant
correspondence in our group-action convention is
\begin{equation}
 \mathcal A(u)=c(u)U_{S(u)}Z,
 \label{eq:ancestor-descendant}
\end{equation}
with $c(u)$ independent of the ancestor variables. Both
formulas follow from the splitting identity for $\psi_i-\bar\psi_i$
and its genus-zero specializations (Kontsevich--Manin,
Getzler, Givental), in the super conventions of
Section~\ref{sec:conv}.

We use \eqref{eq:ancestor-descendant} only on the fixed theories,
with $u$ of positive total Novikov filtration. On such $u$,
$S(u,z)-\id$ has positive filtration, and every Novikov
coefficient has finitely many negative $z$-powers: finitely
many background marks at that coefficient, and $F$'s cotangent
classes are nilpotent. The quantized action is then covered by
Lemma~\ref{lem:finiteness}.

For the master space we abbreviate
$S_W=S_W(0,z)$ and $\mathcal A_W=\mathcal A_W(0)$. Its
loop-variable rationality and its quantized action against the
fixed theories will be established directly by localization in
the next section.

% ==================================================================
\section{Ancestor localization for the master space}\label{sec:loc}
% ==================================================================

The torus action on $W$ splits its Gromov--Witten theory into
contributions from $B$ and $X$. This section proves the resulting
factorization of the master-space ancestor potential. We
abbreviate
\[
 H^{\mathrm{fix}}=H_B\oplus H_X,\qquad
 \eta^{\mathrm{fix}}=\eta_B^t\oplus\eta_X^t.
\]
Restriction identifies this paired space with $H_T^*(W)$ after
inverting $\lambda$. All series use full curve labels
$Q^\beta y^l$.

\begin{proposition}[Ancestor localization]\label{prop:loc-factorization}
There are classes $u_F\in H_F$ of positive Novikov filtration
for $F=B,X$, and a symplectic power series
$R(z)\in\End(H^{\mathrm{fix}})[[z]]$ with $R-\id$ of positive
Novikov filtration, such that, writing
$S_f(z)=S_{F,\mathrm{tw}}(u_F,z)$ and
$S_{\mathrm{bl}}(z)=\bigoplus_F S_f(z)$,
\begin{align}
 S_W(z)&=R(z)S_{\mathrm{bl}}(z),\label{eq:S-factorization}\\
 \mathcal A_W(0)&=c\,U_R
   \prod_{F=B,X}\mathcal A_{F,\mathrm{tw}}(u_F),
   \label{eq:A-factorization}
\end{align}
with $c$ an invertible scalar independent of ancestor variables.
For every curve coefficient, $S_W(z)$ is the expansion of a
rational function of $z$ (at $z=0$ the identity is in Laurent
series). Both $R$ and its inverse admit quantized actions in the
coefficientwise completion used here.
\end{proposition}

\paragraph{Source of the argument.} The same master-space geometry
underlies Fan's reconstruction of projective-bundle invariants;
for ancestors we adapt Coates--Givental--Tseng's localization,
with an additional gluing argument required because the orbit
directions here vary over the positive-dimensional variety $X$.

\subsection{Moving components and their gluing}

A \emph{moving leg} is an irreducible component of a
torus-fixed stable map whose image is not contained in
$W^T=B\sqcup X$. Connected pieces mapping into $B$ or $X$ are
kept as stable-map factors for that component (including their
internal boundary strata).

\begin{lemma}[Leg geometry]\label{lem:leg-geometry}
A moving leg has degree $m\geq 1$ on a line joining
$x\in X$ to $b=\pi(x)\in B$. Its map from $\mathbb P^1$ is
totally ramified at the two endpoints, and its markings and
nodes lie at these endpoints. The leg moduli is the smooth
proper $\mu_m$-gerbe $q_m:\mathfrak L_m\to X$ of $m$th roots of
$\mathcal O_X(-1)$, with endpoint evaluation maps $q_m$ and
$\pi\circ q_m$.

At an endpoint in $F$, the tangent line of the leg has
equivariant first Chern class
\[
        a=\frac{w_F\lambda}{m}+\nu,
\]
with $\nu$ an ordinary degree-two class on $\mathfrak L_m$.
Every node involving a moving leg has nonzero smoothing
character.
\end{lemma}

\begin{proof}
After a finite cover of the torus, the action lifts to the domain
of a fixed stable map. A nontrivial $\C^*$-action on a complete
irreducible curve is trivial on its normalization; since $T$ acts
trivially on $B$ the projection of a moving component is
constant. In a fiber $\mathbb P(E_b\oplus\C)$ the closure of any
nonfixed orbit is the line joining $[0:1]$ to a unique
$[v:0]\in\mathbb P(E_b)$. An equivariant finite map to this
closure is, at the two endpoints, $t\mapsto t^m$; a self-node
would identify distinct fixed points, so the moving component is
smooth and its special points all lie at the endpoints.

Varying the endpoint $[v]$ gives $X$ as the parameter space of
orbit lines. With $q_m:\mathfrak L_m\to X$ the root gerbe of
$\mathcal O_X(-1)$ and $M$ the universal $m$th-root line, the
universal cover
\[
 \mathbb P(M\oplus\mathcal O)\to
 \mathbb P(q_m^*\mathcal O_X(-1)\oplus\mathcal O)\to W,\qquad
 [s:t]\mapsto[s^m:t^m],
\]
is locally the moving leg, and its automorphisms are exactly
the deck group $\mu_m$. This identifies the leg moduli. It is
smooth and proper over the projective variety $X$, and its
ordinary class coincides with its fixed virtual class: on a
given fiber line $L$,
\[
 T_{W/B}|_L\cong\mathcal O(2)\oplus\mathcal O(1)^{\oplus(r-1)},
 \quad
 f^*\pi^*T_B\cong T_{B,b}\otimes\mathcal O_{\mathbb P^1},
\]
so $H^1(\mathbb P^1,f^*T_W)=0$; the fixed part of the ambient
deformation space is exactly $T_{\mathfrak L_m}$.

The universal cover gives tangent classes
$a_X=(\lambda+q_m^*p|_X)/m$ at $X$ and $a_B=-a_X$ at $B$, with
scalar characters $+\lambda/m$ at $X$ and $-\lambda/m$ at $B$. A
node joining a leg to a fixed-map factor has the corresponding
nonzero character. Two legs at a node approach the same fixed
component from opposite signs, so the smoothing character is
$w_F\lambda(1/m+1/m')\ne 0$.
\end{proof}

\subsection{Gluing and virtual localization}

Localization is applied as follows. Distinguish all cut branches
at nodes, then divide by graph automorphisms. Since every
smoothing character at a node cut from a moving leg is nonzero,
fixed deformations cannot smooth those nodes; fixed loci are
described by matching leg endpoints and stable-map pieces to
$B$ or $X$ along evaluation maps. This description holds in
families: degree zero over $B$ forces the projection of a
rational family to factor through its base, and the local
monomial form applies after a trivialization.

\paragraph{Obstruction theory matching.} Normalize pieces
$C_\alpha$ at the cut nodes $q$. The map-deformation complexes
fit into the compatible triangle
\[
 R\Gamma(C,f^*T_W)\to
 \bigoplus_\alpha R\Gamma(C_\alpha,f_\alpha^*T_W)\to
 \bigoplus_q T_{W,f(q)}.
\]
Passing to absolute map deformations adds pointed-domain
deformation and automorphism complexes, and releasing a cut node
adds its smoothing line $T_{q,+}\otimes T_{q,-}$. The ambient
virtual tangent class of the fixed gluing locus is thus
\begin{equation}\label{eq:tangent-gluing}
 [\mathbb T^{\mathrm{vir}}_\Gamma]
 =\sum_\alpha[\mathbb T^{\mathrm{vir}}_\alpha]
  -\sum_q[\ev_q^*T_W]
  +\sum_q[T_{q,+}\otimes T_{q,-}].
\end{equation}
The identity comes from the displayed triangle, so the fixed
part gives the diagonal-matched obstruction theory, not merely
an equality of virtual ranks. On a fixed-map piece the fixed
part is the ordinary theory in $F$; on a leg it is
$T_{\mathfrak L_m}$. The matching terms are $T_F$; every
smoothing line is moving. The fixed virtual class is the
diagonal virtual pullback of these classes, including when two
legs meet or a graph has a cycle.

Cutting a leg off a fixed-map vertex contributes the normal
matching numerator $e_T(N_F)$ and the smoothing denominator
\begin{equation}\label{eq:flag-denominator}
                  \frac1{a-\psi},
\end{equation}
with $\psi$ the cotangent class at the vertex. The numerator
exactly turns the contraction into the inverse-Euler-twisted
$\eta_F^t$. All remaining leg data are cohomology
correspondences independent of the vertex descendants. For an
\emph{external} marked point at the leg endpoint, there is no
smoothing denominator; its cotangent class is $-a$, and
contraction with $\eta_F^t$ again cancels the matching
numerator.

\subsection{Nilpotent denominators and dummy derivatives}

The ordinary class $\nu$ in $a$ need not descend to $F$, so we
expand in $\mathfrak L_m$ before pushing forward. If $a=a_0+\nu$
with $a_0=w_F\lambda/m\ne 0$, then
\begin{equation}\label{eq:nilpotent-denominator}
 \frac{1}{a-s}=\sum_{j\geq 0}\frac{(-\nu)^j}{(a_0-s)^{j+1}}
\end{equation}
(finite sum). Let $C$ be a correspondence coefficient on a cut
piece with endpoint map $e$ to $F$, and put
$h_j=e_*(\nu^j C)$ (with virtual and Euler factors included).
For any flag function $K$, define
\begin{equation}\label{eq:dummy-derivatives}
 K(a)h:=\sum_{j\geq 0}\frac{1}{j!}\,
        \partial_\alpha^j K(\alpha)\big|_{\alpha=a_0}\,h_j,
\end{equation}
a \emph{dummy} scalar $\alpha$: the derivative acts only on $K$.
For example $S_f(a)h/(a+z)$ means \eqref{eq:dummy-derivatives}
with $K(\alpha)=S_f(\alpha)/(\alpha+z)$. At several flags use
joint moments and independent dummy scalars. A pure ordinary
cohomology vector is the special case with only $h_0$.

\paragraph{Coinvariance and finiteness.} All graph sums below are
coefficientwise finite: each moving leg has positive tautological
degree, so its multiplicity and the number of legs is bounded at
fixed ample degree; there are finitely many splittings into
effective full curve labels; stability bounds remaining degree
zero pieces once genus and distinguished markings are fixed.

\subsection{The genus-zero factor $R$}

Fix $F\in\{B,X\}$. An \emph{end} at $F$ is a genus-zero unmarked
tree attached to a fixed-map vertex by a moving leg, with the
vertex itself omitted. Its contribution, including the
$(a-s)^{-1}$ denominator at the attachment, is a cohomology-valued
power series $\epsilon_F(s)$ of positive Novikov filtration. A
\emph{tail} has one external primary insertion $b$ and contributes
$h_b/(a-s)$; the moments $h_b$ absorb curve monomials, stack and
symmetry factors, and the linear dependence on $b$.

End denominators have infinite Taylor expansions; in identities
involving these backgrounds we use the completion
$\widehat\cH_+=H_F[[z]]$ of the positive space, with localized
Novikov coefficients, and $[\,\cdot\,]_+$ the nonnegative part.
The usual cone identities extend coefficientwise (positive
filtration bounds the number of insertions; nilpotence of
cotangent classes bounds indices).

Let $u_F$ be the primary parameter of the tangent space to the
genus-zero cone of the $F,\mathrm{tw}$ theory at background
$\epsilon_F$; equivalently it is determined by
\begin{equation}\label{eq:u-equation}
 x_F(z):=[S_f(z)(\epsilon_F(z)-u_F)]_+\in z\widehat\cH_+,\quad
 S_f(z)=S_{F,\mathrm{tw}}(u_F,z).
\end{equation}
Applying $S_f$ makes the cone point belong to
$z\widehat\cH_+$, and $[S_f(z)(-z+u_F)]_+=-z$; the linearization
of the constant term is $-\id$ at zero Novikov degree, so $u_F$
exists uniquely and has positive filtration.

The tangent space being the Hessian graph implies that its
two-point function depends on $\epsilon_F$ only through $u_F$.
The genus-zero two-point identity (dimension-counted from
topological recursion) is
\begin{equation}\label{eq:two-point}
 \frac{\eta_F^t(b,c)}{w+z}
 +\Bigl\langle
    \frac{b}{w-\psi},\frac{c}{z-\psi}
 \Bigr\rangle_{0,\epsilon_F}^{F,\mathrm{tw}}
 =\frac{\eta_F^t(S_f(w)b,S_f(z)c)}{w+z}.
\end{equation}
It is rational in $w,z$ at each curve coefficient.

Localize the master matrix element $\eta(b,S_W(z)c)$ for $c$
supported on $F$. If the input lies on a fixed-map vertex, the
unmarked outgoing trees provide the background $\epsilon_F$; the
insertion $b$ is either on that vertex ($b|_F$) or on a tail with
attachment $h_b/(a-\psi)$. If $b$ is a marked leg endpoint, the
endpoint rule supplies the metric term in \eqref{eq:two-point},
with denominator $a+z$. Taking the $w^{-1}$ coefficient of
\eqref{eq:two-point} handles a primary slot, and we get
\begin{equation}\label{eq:S-tail}
 \eta(b,S_W(z)c)=\eta_F^t(b|_F,S_f(z)c)
   +\sum_{\text{tails}}
      \frac{\eta_F^t(S_f(a)h_b,S_f(z)c)}{a+z},
\end{equation}
giving
\begin{equation}\label{eq:R-tail}
 (R(z)^*b)_F=b|_F+
   \sum_{\text{tails}}\frac{S_f(a)h_b}{a+z}
\end{equation}
as a power series at $z=0$. Consequently $R$ is upper triangular
and
\begin{equation}\label{eq:R-symplectic}
 R(-z)^*R(z)=\id,
\end{equation}
since both $S_W$ and $S_{\mathrm{bl}}$ are symplectic and their
symbolic identities are rational at each curve coefficient.
Every nonidentity term in \eqref{eq:R-tail} has a moving leg, so
$R-\id$ has positive Novikov filtration.

\paragraph{Translation and bridge.} The one-point function is
$J_W(-z)=-zS_W(z)^{-1}1$. The string equation identifies it as
\[
 J_W(-z)=-z1+
 \sum_{\beta,l}Q^\beta y^l(\ev_1)_*
 \frac{[\overline{\mathcal M}_{0,1}(W,(\beta,l))]^{\mathrm{vir}}}
      {-z-\psi_1},
\]
where only stable one-marked terms occur. Its regular part on
$F$ is $-z+\epsilon_F(z)$ (a marked leg endpoint contributes an
end; markings on the fixed vertex contribute only negative
powers). Hence $\epsilon_F(z)=z+[J_W(-z)|_F]_+$. Using $S_f$'s
nonpositive powers, $[S_f(z)z]_+=z+u_F$, and the factorized
identity gives
\begin{equation}\label{eq:translation}
             \bigoplus_F x_F(z)=z(1-R(z)^{-1}1).
\end{equation}

A \emph{bridge} is an unmarked genus-zero tree with two fixed-map
attachments and moving legs at both; its contribution, dressed
by $S_f(a),S_{f'}(a')$ on the two correspondence slots, is
$D(s,s')$. Localizing the master two-point function and using
`path determines unique bridge' gives
\begin{equation}\label{eq:bridge-two-point}
 \frac{S_W(w)^*S_W(z)}{w+z}
 =S_{\mathrm{bl}}(w)^*
  \Bigl(\frac{\id}{w+z}+D(-w,-z)\Bigr)
  S_{\mathrm{bl}}(z),
\end{equation}
so, with \eqref{eq:S-factorization} and \eqref{eq:R-symplectic},
\begin{equation}\label{eq:bridge-kernel}
 D(-w,-z)=\frac{R(w)^*R(z)-\id}{w+z}.
\end{equation}
The numerator vanishes at $w=-z$, so the right side is a power
series at $(w,z)=(0,0)$, as is the dressed bridge expression; the
identity is first proved for symbols and then expanded as a
Taylor series.

\subsection{From descendants at flags to ancestors}

Two ancestor identities convert surviving vertices to ancestor
theory. For a fixed target with pairing $\eta^t$, let
$\mathcal A(\epsilon;t)$ denote the mixed potential with
distinguished descendants $t(\bar\psi)$ and additional insertions
$\epsilon(\psi)$; the ancestors forget the additional marks.
Parameter $\epsilon$ of positive filtration; $u$ and $x(z)$ as
in \eqref{eq:u-equation}.

\begin{lemma}[Vertex conversion]\label{lem:vertex-conversion}
In summed vertex correlators with background $\epsilon$, an
insertion at a distinguished flag transforms as
\begin{equation}\label{eq:flag-dressing}
 \frac{h}{a-\psi}\longmapsto\frac{S(u,a)h}{a-\bar\psi},
\end{equation}
also with additional ancestor-class powers at the flag and with
further distinguished flags (all flag evaluation classes in $h$
before applying $S(u,a)$). Moreover, up to a scalar independent
of $t$,
\begin{equation}\label{eq:mixed-potential}
              \mathcal A(\epsilon;t)=\mathcal A(u;t+x).
\end{equation}
\end{lemma}

\begin{proof}
These are the ancestor identities of Getzler and
Kontsevich--Manin, in the form used by Coates--Givental--Tseng;
we spell out the arguments because the background has
descendants.

The identity
$\frac{1}{a-\psi}=\frac{1}{a-\bar\psi}
+\frac{\psi-\bar\psi}{(a-\psi)(a-\bar\psi)}$
reduces \eqref{eq:flag-dressing} to the splitting divisor of
$\psi-\bar\psi$. That divisor separates a genus-zero twig at the
mark, a subset of background marks, and an attaching node.
Splitting the virtual class contracts the twig's two-point
function with the rest of the vertex. Summing all twigs plus the
direct term gives $S(u,a)h$ by the primary-slot instance of
\eqref{eq:two-point}. Ancestor classes at the original markings
are pulled back from the stabilized curve, so they do not enter
the twig calculation. For a tangent class with nilpotent part,
apply the scalar identity then the finite dummy derivatives of
\eqref{eq:dummy-derivatives}.

For \eqref{eq:mixed-potential}, write $\epsilon=u+(\epsilon-u)$
and expand multilinearly. Forget the primary $u$ marks alone; the
splitting argument converts the remaining descendants to
$x(\bar\psi)=[S(u,z)(\epsilon(z)-u)]_+|_{z=\bar\psi}$. The
original ancestors still forget the $x$ marks; the new ones
retain them. Their cotangent-class difference has a boundary
divisor with a rational tail carrying one original mark and
$k\geq 1$ marks to be forgotten; its stable-curve factor is
$\overline{\mathcal M}_{0,k+2}$ of dimension $k-1$; every $x$
insertion has positive ancestor power (since $x\in
z\widehat\cH_+$), so the product restricts to at least degree
$k$ and vanishes. Applying this at every distinguished flag
replaces original ancestors by those retaining all $x$ marks;
multilinearity gives \eqref{eq:mixed-potential}. Terms that only
become stable after adding $x$ contribute only a scalar (in
genus zero, positive ancestor powers exceed the stable-curve
dimension; in genus one, no original distinguished mark leaves a
scalar).
\end{proof}

\subsection{Assembly}

Fix a stable map contributing to a stable ancestor invariant of
$W$. Forget the map and stabilize the distinguished curve. Every
moving leg disappears (it has at most two special points).
Prune unmarked rational trees and suppress rational chains with
two remaining flags. Surviving fixed-map factors are kept with
their own stabilization and internal boundary. Removed pieces
between surviving vertices are \emph{bridges}; removed pieces
ending in a distinguished marking are \emph{tails}; all others
are \emph{ends}.

Ends give the background $\epsilon_F$. Lemma~\ref{lem:vertex-conversion}
converts bridge and tail denominators to ancestor denominators and
inserts the factors $S_f(a)$. Bridges become exactly
$D(\bar\psi,\bar\psi')$. Direct markings together with all tails
become, using \eqref{eq:R-tail},
\[
 R(-z)^*t(z)=R(z)^{-1}t(z),\qquad z=\bar\psi.
\]
Finally \eqref{eq:mixed-potential} replaces $\epsilon$ by $u_F$
and adds $x_F$. Combined with \eqref{eq:translation} the vertex
substitution is
\begin{equation}\label{eq:Wick-substitution}
 t(z)\longmapsto R(z)^{-1}t(z)+z(1-R(z)^{-1}1).
\end{equation}

Quantize this substitution. If
$D(s,s')=\sum_{i,j\geq 0}D_{ij}s^i(s')^j$, the upper-triangular
contraction operator is $\hbar/2$-times the second derivative
with coefficients $D_{ij}$, using the categorical inverse of
$\eta^{\mathrm{tw}}$. Exponentiating it on the product of fixed
ancestors and applying \eqref{eq:Wick-substitution} is exactly
$U_R$: the kernel is \eqref{eq:bridge-kernel}, and the signs
$-w,-z$ come from the negative Darboux modes $(-z)^{-i-1}$. The
graph sum reproduces this: labeling flags first and then dividing
by permutations produces the same factorials and graph
automorphisms; a bridge or self-bridge contributes one factor of
$\hbar$; super signs and permutations are exactly those of the
tensor rules.

Components without distinguished markings may multiply
$\mathcal A_W$ by an invertible scalar; this is the $c$ in
\eqref{eq:A-factorization}. The invertibility of $U_R$ is
coefficientwise: $R-\id$ has positive filtration so its inverse
and logarithm are defined degree by degree; at each fixed
coefficient the number of contributing filtration-positive
factors is finite; ancestor powers are bounded by $3g-3+n$; fixed
descendant identities are bounded by fixed-target stable-map
dimensions. This completes the proof of
Proposition~\ref{prop:loc-factorization}.

% ==================================================================
\section{Grading and the Iritani shift}\label{sec:shift}
% ==================================================================

We now build two commuting operators on the equivariant quantum
cohomology of $W$: the calibrated grading $G_a$, and Iritani's
shift $T_a$, which translates $\lambda\mapsto\lambda+z$. The
$T_a$ admits a simple closed formula on each fixed component, and
its spectrum on fiber classes has $r+1$ branches that will
organize the transfer.

Throughout $\star$ means the small equivariant quantum product of
$W$ at primary background zero. Put
\[
 c=c_1^T(TW),\qquad \kappa=c_1(TB)+c_1(E).
\]
The equivariant projective-bundle formula gives
\begin{equation}
 c=(r+1)p+\pi^*\kappa+\lambda,\qquad
 \int_{(\beta,l)}c_1(TW)=(r+1)l+\int_\beta\kappa.
 \label{eq:c1}
\end{equation}
Choose a homogeneous Hodge basis of $H^*(B)$ and the resulting
polynomial basis $\pi^*b\,p^j$ ($0\leq j\leq r$) of $H_T^*(W)$
over $\C[\lambda]$. In this basis, $\mu$ acts by the first Hodge
degree minus $\dim_\C(W)/2$, and does not differentiate the
$\lambda$-coefficient.

\subsection{The calibrated grading}

Define
\begin{equation}
 G_d=z\partial_z+\lambda\partial_\lambda+\tfrac12+\mu+
      \frac{c\,\cup}{z},\quad
 G_a=z\partial_z+\lambda\partial_\lambda+\tfrac12+\mu+
      \frac{c\star}{z}.
 \label{eq:gradings}
\end{equation}
The subscripts distinguish the constant (descendant) frame
from the quantum (ancestor) frame. Both differentiate $\lambda$
and hold Novikov variables fixed.

\begin{lemma}\label{lem:grading}
With $S_W=S_W(0,z)$ in the conventions of
Section~\ref{sec:conv},
\[
                    G_a=S_W G_d S_W^{-1}.
\]
\end{lemma}

\begin{proof}
Let $D_c$ be the derivation of the Novikov ring sending
$Q^\beta y^l$ to $((r+1)l+\int_\beta\kappa)Q^\beta y^l$.
Homogeneity of the two-point invariants defining $S_W$ says
\begin{equation}
 (z\partial_z+\lambda\partial_\lambda+D_c)S_W
   =S_W\mu-\mu S_W.
 \label{eq:homogeneity}
\end{equation}
This uses the first Hodge degree: evaluation maps, virtual class,
and cotangent classes are algebraic, so their push-pull has the
Hodge bidegree prescribed by virtual dimension. Concretely, if
the input and output basis vectors have first Hodge degrees
$h_j,h_i$, the coefficient of $Q^\beta y^l z^{-k-1}$ in
$S_W$'s corresponding entry has $\lambda$-degree
$h_j+k+1-h_i-((r+1)l+\int_\beta\kappa)$, since $\lambda$ and
$z$ are both bidegree $(1,1)$. This is \eqref{eq:homogeneity}.
It applies equally to primitive and odd inputs.

The divisor equation gives
\begin{equation}
 zD_c S_W=(c\star)S_W-S_W(c\cup).
 \label{eq:divisor}
\end{equation}
Substituting into \eqref{eq:homogeneity} gives $G_a S_W=S_W G_d$.
The identities first hold in the $z=\infty$ expansion; by
coefficientwise rationality of $S_W$ (from localization) they
also hold in the $z=0$ Laurent expansion.
\end{proof}

\subsection{The Iritani shift}

\paragraph{Iritani's input.} Let $Y$ be a smooth projective
variety with a $\C^*$-action, form its associated bundle
$\widehat Y\to\mathbb P^1$, and let $\sigma_{\min}$ be the
section class of the fixed component with positive normal
weights. The operator defined by two-fiber section invariants,
with monomials $Q^{\widehat d-\sigma_{\min}}$, is an unlocalized
equivariant cohomology operator. After reversing the sign of
Iritani's loop variable, it translates
$\lambda\mapsto\lambda+z$ and satisfies
\begin{equation}
 T_a=S_Y T_d S_Y^{-1},\qquad
 T_d\big|_F
 =Q^{\sigma_F-\sigma_{\min}}
  \prod_{m,\,x}
  \frac{\prod_{a=-\infty}^{0}(x+m\lambda-az)}
       {\prod_{a=-\infty}^{-m}(x+m\lambda-az)}
  e^{z\partial_\lambda}.
 \label{eq:iritani}
\end{equation}
Here $x$ runs through the ordinary Chern roots of the
weight-$m$ normal subbundle of $F$; each quotient has only
finitely many remaining factors. This is the projective
specialization of Iritani's shift theorem; the source's
seminegativity hypothesis is automatic for a complete target,
and its calibration is related to ours by
$S_Y(z)=M(-z)^{-1}$.

\paragraph{Associated geometry.} For the weight-one action on
$\mathcal O_B$, the associated space is
\begin{equation}
 \widehat W
  =\mathbb P_{B\times\mathbb P^1}
   \Bigl(\pr_B^*E\oplus\pr_{\mathbb P^1}^*
   \mathcal O(-1)\Bigr)
  \cong W\times(\C^2\setminus\{0\})/\C^*,
 \label{eq:associated-space}
\end{equation}
where $s\cdot(w,v_1,v_2)=(s\cdot w,s^{-1}v_1,s^{-1}v_2)$.
An additional torus scales $v_2$; its equivariant parameter is
Iritani's loop variable before sign reversal. The fiber over
$[1:0]$ carries the original action and the fiber over $[0:1]$
carries the action composed with the additional torus; identifying
their cohomologies via the induced change of torus variables is
the translation in \eqref{eq:iritani}.

Write $\zeta$ for that parameter before sign reversal, and put
$\rho_0(t,u)w=t\cdot w$, $\rho_1(t,u)w=(tu)\cdot w$. The change
of torus variables induces
$\Phi_1:H^*_{T\times\C^*,\rho_0}(W)\to H^*_{T\times\C^*,\rho_1}(W)$
with
\[
 \Phi_1(f(\lambda,\zeta)a)=f(\lambda+\zeta,\zeta)\Phi_1(a).
\]
Let $\iota_0,\iota_\infty$ denote the fiber inclusions. The
section correspondence is defined by
\begin{equation}
 (\widetilde T a,b)_\infty
  =\sum_{\beta,l}Q^\beta y^l
   \bigl\langle\iota_{0*}a,\iota_{\infty*}b
   \bigr\rangle^{\widehat W,T\times\C^*}_{0,2,
   \sigma_{\min}+(\beta,l)},
 \label{eq:shift-section}
\end{equation}
with only effective section classes included. Define
$T_a=(\Phi_1^{-1}\widetilde T)|_{\zeta=-z}$; semilinearity
translates $\lambda\mapsto\lambda+z$.

\paragraph{Section classes.} At $X$ the normal weight is positive,
so $X$ gives $\sigma_{\min}$. A section obtained from a point of
$X$ has tautological degree zero in \eqref{eq:associated-space};
the section from the trivial-summand fixed component $B$ is the
line $\mathcal O(-1)$ and has tautological degree one. Both have
base class zero, so $\sigma_B-\sigma_{\min}$ is the fiber-line
class of $W$.

The curve labels fit: on $H_2(W;\Z)$ we have $(\beta,l)$; on
$H_2(\widehat W;\Z)$ a section class is $(\beta,1,l)$;
subtracting $\sigma_{\min}=(0,1,0)$ leaves $(\beta,0,l)$. The
fiber inclusions preserve $\beta$ and tautological degree, so the
two fibers share their integral identification. Splittings of a
section map add integral curve classes; \eqref{eq:iritani} does
not collapse classes to numerical equivalence.

\begin{proposition}\label{prop:shift}
There is a shift operator
\[
 T_a=A(z,\lambda,Q,y)e^{z\partial_\lambda}
\]
on the small quantum cohomology module of $W$, whose matrix $A$
is polynomial in $z,\lambda$ at each curve coefficient in the
equivariant basis. Its calibration is
\begin{equation}
 T_a=S_W T_d S_W^{-1},
 \label{eq:shift-calibration}
\end{equation}
where, writing $v=x+w_F\lambda$ for the equivariant normal roots,
\begin{equation}
 T_d\big|_F
   =y^{j_F}
    \prod_x
    \begin{cases}
     x+\lambda, & F=X,\\[2pt]
     (x-\lambda-z)^{-1}, & F=B
    \end{cases}
    e^{z\partial_\lambda},\quad
    j_X=0,\; j_B=1.
 \label{eq:shift-fixed}
\end{equation}
All operators act on the full equivariant cohomology with its
super pairing.
\end{proposition}

\begin{proof}
$W$ is smooth and projective, and \eqref{eq:associated-space} is
projective, so its section invariants are defined by proper
equivariant pushforward. The two fiber insertions are polynomial
equivariant classes, and the equivariant pairing of $W$ is
perfect over $\C[\lambda]$, so the operator is polynomial in both
equivariant parameters at each curve coefficient. Normal weights
are $1$ at $X$ and $-1$ at $B$; in \eqref{eq:iritani} they give
the factors $v$ and $(v-z)^{-1}$ respectively, and the section
calculation gives $y^{j_F}$. In our fundamental-solution
convention $S_W(z)=M(z)^*=M(-z)^{-1}$, so Iritani's identity
$M T_a=T_d M$, with his $z$ replaced by $-z$, is exactly
\eqref{eq:shift-calibration}.
\end{proof}

\begin{lemma}\label{lem:polynomiality}
For each fixed base class $\beta$, the matrix coefficients of
$A$ and of $c\star$ are polynomials in $y,z,\lambda$ (with no
$z$-dependence in $c\star$). In particular they are holomorphic
in $y$ before any localization in $\lambda$.
\end{lemma}

\begin{proof}
The dual of the bundle in \eqref{eq:associated-space} is the sum
of the globally generated bundles $\pr_B^*E^*$ and
$\pr_{\mathbb P^1}^*\mathcal O(1)$, so its tautological line
bundle is globally generated and every effective section class
has $l\geq 0$; subtracting $\sigma_{\min}$ does not change $l$.

Fix basis vectors of an input and a paired output. The
projective-bundle formula on $\widehat W$ gives
\begin{equation}
 \operatorname{vdim}_\C
 \overline{\mathcal M}_{0,2}(\widehat W,
 \sigma_{\min}+(\beta,l))
 =\dim_\C W+1+(r+1)l+\int_\beta\kappa,
 \label{eq:section-dimension}
\end{equation}
so virtual dimension increases by $r+1$ when $l$ increases by one.
Proper equivariant integration is polynomial in parameters. If
virtual dimension exceeds total insertion degree, the result has
negative cohomological degree and vanishes, so only finitely many
$l$ occur for this entry and $\beta$. The same argument with the
three primary insertions defining a quantum product proves it for
$c\star$.
\end{proof}

\begin{lemma}\label{lem:commutation}
$[G_a,T_a]=0$.
\end{lemma}

\begin{proof}
By the two calibration identities it suffices to work in the
descendant frame. Equivariant homogeneity of restriction gives
$\lambda\partial_\lambda+\mu=
\lambda\partial_\lambda+\mu_F-n_F/2$ on the $F$-summand, and
$c|_F=c_1(TF)+\sum_x(x+w_F\lambda)$. Write
$T_d|_F=a_F E_z$ with $E_z=e^{z\partial_\lambda}$. The
combination $z\partial_z+\lambda\partial_\lambda$ commutes with
$E_z$; combined with first-Hodge-degree counting it applies $1$
to each $x+\lambda$ factor and $-1$ to each $(x-\lambda-z)^{-1}$
factor, so its commutator with $a_F$ is $W_F a_F$, where
$W_F=n_Fw_F$ (with $y^{j_F}$ held fixed). On the other hand
$E_z(c|_F)=(c|_F+W_F z)E_z$ and
$[c|_F/z,a_FE_z]=-W_F a_FE_z$; the two contributions cancel.
Scalar-centering and cup-multiplication terms introduce no further
commutators.
\end{proof}

\subsection{The fiber spectrum of the shift}

Let $Q^{>0}$ be the ideal of positive base degree. Since an
effective nonzero base class has positive ample degree, $Q^{>0}$
retains exactly the maps with constant projection to $B$. The
$y$-variable is retained.

\begin{lemma}\label{lem:fiber-operator}
On $H_T^*(W)$ with the fiber quantum product,
\begin{equation}
 T_a\bmod(z,Q^{>0})=(p+\lambda)\star.
 \label{eq:seidel}
\end{equation}
\end{lemma}

\begin{proof}
The fiber quantum product is linear over $H^*(B)$ (all
genus-zero fiber evaluations have the same projection to $B$,
so base classes pull out of pushforward), and $S_W$ in the
reduction is likewise $H^*(B)$-linear. So $T_a$ is
$H^*(B)$-linear.

Write $D_y=y\partial_y$. The divisor equation gives
$S_W(zD_y-p\cup)S_W^{-1}=zD_y-p\star$, and the left operator
commutes with $T_d$. On $X$, the multiplier has no $y$ factor
and $p|_X$ is independent of $\lambda$. On $B$, $p|_B=-\lambda$
and $[zD_y,T_d|_B]=zT_d|_B$, $[\lambda,T_d|_B]=-zT_d|_B$, so
the two terms cancel there too. Reducing modulo $z$ proves that
$T_a|_{z=0}$ commutes with $p\star$.

Assign cohomological degree one to $p,\lambda,z$ and degree
$r+1$ to $y$. Polynomiality in $\lambda$ forces $p\star p^k$
to equal the classical power for $k\leq r$; these powers form
an $H^*(B)[\lambda]$-basis, so an $H^*(B)[\lambda]$-linear
operator commuting with $p\star$ is determined by its value on
$1$. In this grading $T_a$ has degree one (equivalently,
\eqref{eq:section-dimension} at $\beta=0$ is
$\dim W+1+(r+1)l$, and each fiber inclusion raises insertion
degree by one); so $T_a(1)|_{z=0}$ has no positive-$y$
coefficient. At $y=0$ we have $S_W=1$, and
\eqref{eq:shift-fixed} applied to $1$ restricts to $p+\lambda$
on $X$ and to zero on $B$; these are the restrictions of
$p+\lambda$ to the fixed locus. Thus $T_a(1)|_{z=0}=p+\lambda$,
and \eqref{eq:seidel} follows.
\end{proof}

Let $I=H^{>0}(B;\C)$; it is nilpotent, also in the presence of
odd classes. Reducing the fiber quantum algebra modulo $I$ gives
\begin{equation}
 \C[\lambda,y,p]\big/\bigl(p^r(p+\lambda)-y\bigr).
 \label{eq:fiber-relation}
\end{equation}
Indeed, the classical relation reduces to $p^r(p+\lambda)=0$; in
degree $r+1$ the only possible positive curve correction is a
constant multiple of $y$. Its coefficient is the degree-one
fiber line invariant:
\[
 \Bigl\langle H,H^r,H^r\Bigr\rangle^{\mathbb P^r}_{0,3,\mathrm{line}}
 =1,
\]
since two general point conditions determine a unique line and
the remaining marked point meets a general hyperplane in one
point; convexity makes this the virtual count as well. The
coefficient has degree zero, so may be computed at $\lambda=0$.

\begin{proposition}\label{prop:spectrum}
For generic $(y,\lambda)$, the distinct eigenvalue branches of
$T_a\bmod(z,Q^{>0})$ are
\begin{equation}
 \lambda+h,\qquad h^r(h+\lambda)=y.
 \label{eq:spectrum}
\end{equation}
Each branch may have nonzero multiplicity and nilpotent part. For
$\lambda\ne 0$, near $y=0$ one branch tends to $0$ and a cluster
of $r$ branches tends to $\lambda$.
\end{proposition}

\begin{proof}
On $H^*(B)[\lambda,y,p]/(I)$, Lemmas~\ref{lem:fiber-operator}
and \eqref{eq:fiber-relation} give the eigenvalues. The finite
filtration by powers of $I$ is preserved, and each graded piece
of the induced operator is the same fiber multiplication tensored
with $I^m/I^{m+1}$, so the characteristic polynomial has the same
set of roots at each step. Away from the critical values of
$h\mapsto h^r(h+\lambda)$, roots are distinct; at $y=0$ they are
$-\lambda$ (multiplicity one) and $0$ (multiplicity $r$).
Translating by $\lambda$ gives the final assertion. The argument
uses only nilpotence of $I$, not semisimplicity of $B$'s quantum
cohomology.
\end{proof}

Polynomial dependence of $G_a,T_a$ and finiteness of the spectral
branches are what the next section needs. Both calibration
identities kept full curve labels, so returning to $y=0$ will
still compare the individual descendant theories on $B$ and $X$.

% ==================================================================
\section{Spectral projections and block gradings}\label{sec:spec}
% ==================================================================

The shift $T_a$ has $r+1$ spectral branches at generic fiber
parameter, while localization distinguishes the two fixed
manifolds $B$ and $X$. We construct operators on individual
spectral blocks and compare their sums at $y=0$. This section is
purely classical: it builds and manipulates operators (no
quantized constraints yet).

\subsection{Functional calculus for differential series}

Fix a finite-dimensional vector space $V$ underlying an
equivariant basis of $H_T^*(W)$. On a small open $U$ in the
$(y,\lambda)$-plane, let $\mathcal O_U$ be holomorphic functions
and set
\[
 \mathscr D_\lambda(U)
 =\Bigl\{\sum_{j=0}^J a_j(y,\lambda)\partial_\lambda^j:
   J<\infty,\ a_j\in\mathcal O_U\Bigr\}.
\]
The multiplication is composition, so
$\partial_\lambda a=a\partial_\lambda
+\partial_\lambda(a)$; there is no $y$-differentiation. Let
$\Lambda_B$ be the base Novikov completion and
$\Lambda_{B,+}$ its positive-degree ideal. We use the completed
algebra
\begin{equation}
 \mathscr A_U
 =\Bigl\{\sum_{\beta,m\geq 0}Q^\beta z^m A_{\beta,m}:
   A_{\beta,m}\in\End(V)\otimes\mathscr D_\lambda(U)\Bigr\}.
 \label{eq:spec-algebra}
\end{equation}
Completion is by $z$ and ample base degree. Each $A_{\beta,m}$
has finite differential order, but the order may grow with
$\beta,m$. Products at a fixed coefficient are finite by Novikov
finiteness. The element $z$ is central, and $z\partial_z$ is
adjoined by
$[z\partial_z,Q^\beta z^m A_{\beta,m}]
=mQ^\beta z^mA_{\beta,m}$.

Suppose $T\in\mathscr A_U$ has order-zero reduction
\[
 T^{(0)}=T\bmod(z,\Lambda_{B,+})
       \in\End(V)\otimes\mathcal O_U.
\]
After shrinking $U$, fix contours in the complex $\xi$-plane
avoiding $\Spec T^{(0)}$ and enclosing specified eigenvalue
groups; keep contours fixed as $y,\lambda$ vary. Set
$R_0(\xi)=(\xi-T^{(0)})^{-1}$ and $K=T-T^{(0)}$. The formal
resolvent is
\begin{equation}
 R_T(\xi)=(\xi-T)^{-1}
    =\sum_{n\geq 0}(R_0(\xi)K)^n R_0(\xi).
 \label{eq:resolvent}
\end{equation}
Each factor $K$ raises the joint $z$/base filtration, so at each
coefficient this is a finite matrix sum, meromorphic in $\xi$
with poles exactly at $\Spec T^{(0)}$.

\begin{lemma}[Coefficientwise holomorphic calculus]\label{lem:calculus}
For $T$ as above and a scalar holomorphic $f$ on a neighborhood
of $\Spec T^{(0)}$ (the neighborhood may be disconnected), the
contour integral
\begin{equation}
 f(T)=\frac{1}{2\pi i}\int_\Gamma f(\xi)R_T(\xi)\,d\xi
 \label{eq:functional-calculus}
\end{equation}
is unital, multiplicative, and compatible with holomorphic
composition: $g(f(T))=(g\circ f)(T)$ when the functions are
defined on the indicated neighborhoods. It gives mutually
orthogonal idempotents summing to the identity (via indicator
functions of spectral groups), all commuting with $T$.

If additionally $[T,\lambda]=zT$, then
\begin{equation}
 [f(T),\lambda]=z\bigl(\xi f'(\xi)\bigr)(T).
 \label{eq:functional-commutator}
\end{equation}
If a differential operator $G$ commutes with $T$, acts
continuously on the coefficientwise Laurent extension of
\eqref{eq:spec-algebra}, and commutes with the scalar spectral
variable $\xi$, then $[G,f(T)]=0$.
\end{lemma}

\begin{proof}
The inverse identities for \eqref{eq:resolvent} give the
resolvent relation
$R_T(\xi)R_T(\eta)=(R_T(\xi)-R_T(\eta))/(\eta-\xi)$ (both
spectral variables central). Nested-contour integration and
scalar Cauchy give multiplicativity. Unitality follows directly
from \eqref{eq:resolvent}. For the composition rule, apply the
calculus to $(\eta-f(\xi))^{-1}$ at $\eta$ outside $\Spec f(T^{(0)})$:
the product with $\eta-f(T)$ is the identity, so it is the
resolvent of $f(T)$; integrating against $g(\eta)$ gives
$g(f(T))=(g\circ f)(T)$.

For \eqref{eq:functional-commutator}, use the inverse rule:
$[R_T(\xi),\lambda]=R_T(\xi)[T,\lambda]R_T(\xi)
=z(\xi R_T(\xi)^2-R_T(\xi))=-z\partial_\xi(\xi R_T(\xi))$.
Integration by parts gives \eqref{eq:functional-commutator}.
Finally, $[G,T]=[G,\xi]=0$ implies $[G,R_T(\xi)]=0$ by the
inverse identity; integration gives $[G,f(T)]=0$. Contours
held fixed locally allow coefficientwise differentiation under
the integral.
\end{proof}

We will need a larger algebra for frame changes:
$\mathcal O$ a ring of parameter germs stable under
$\partial_\lambda$, $\mathcal M$ the base Novikov monoid or the
full monoid $\{(\beta,l): l\geq 0\}$:
\begin{equation}
 \mathscr A^{\mathrm{Laur}}_{\mathcal M}(\mathcal O)
 =\Bigl\{\sum_{d\in\mathcal M}q^d
   \sum_{m\geq m(d)}z^m A_{d,m}:
   A_{d,m}\in\End(V)\otimes\mathscr D_\lambda(\mathcal O),\
   m(d)\in\Z\Bigr\}.
 \label{eq:laurent}
\end{equation}
Here $q^d$ means $Q^\beta$ or $Q^\beta y^l$. Multiplication is
well-defined: each fixed $d$ has finitely many decompositions,
and the two Laurent lower bounds leave finitely many $z$-summands
at a fixed power.

\begin{lemma}[Change of frame]\label{lem:gauge}
Let $T,T'$ have coefficientwise resolvents on a common contour
as in Lemma~\ref{lem:calculus}, both in
\eqref{eq:laurent}. Suppose $M,M^{-1}$ lie in the same algebra,
do not depend on $\xi$, and $T'=MTM^{-1}$. Then
\[
 R_{T'}(\xi)=MR_T(\xi)M^{-1},\qquad
 f(T')=Mf(T)M^{-1}
\]
for every $f$ from the contour calculus.
\end{lemma}

\begin{proof}
$MR_T(\xi)M^{-1}$ is a two-sided inverse of $\xi-T'$; integrate
coefficientwise and use the Laurent lower bounds to keep products
finite at each coefficient.
\end{proof}

Two forms of $M$ will occur. First, an identity-modulo-positive-
Novikov matrix whose coefficients are Laurent expansions at $z=0$
of rational functions of $z$ (its Novikov inverse shares this
property). Second, $\exp(N/z)M_+(z)$ with $N$ a nilpotent
cup-product and $M_+(z),M_+^{-1}$ regular at $z=0$; both forms
satisfy the Laurent requirement.

\subsection{Block operators without $\lambda$-differentiation}

We apply the calculus to $T=T_a$. By Lemma~\ref{lem:polynomiality},
$T_a$'s coefficients are polynomial in $y,\lambda,z$ at each fixed
base degree (before the shift $\exp(z\partial_\lambda)$ is
expanded), so $T_a\in\mathscr A_U$. By Proposition~\ref{prop:spectrum},
the spectrum of the reduction $T^{(0)}_a$ is
\begin{equation}
 a_i=\lambda+h_i,\qquad h_i^r(h_i+\lambda)=y.
 \label{eq:spec-eigenvalues}
\end{equation}
Take $\lambda\neq 0$, $y\neq 0$, exclude roots coinciding, and
shrink to a small open $U$. Choose
logarithms near each $a_i$ (a whole generalized eigenspace; no
diagonalizability assumed). Define
\begin{equation}
 P_i=\frac{1}{2\pi i}\int_{\gamma_i}R_{T_a}(\xi)\,d\xi,\quad
 \mathscr L_i=\frac{1}{2\pi i}\int_{\gamma_i}
   \log\xi\, R_{T_a}(\xi)\,d\xi,
 \label{eq:projectors}
\end{equation}
where $\gamma_i$ encloses $a_i$ with its multiplicity. For a
group of eigenvalues use the sum of contours.

\begin{proposition}[Removal of coefficient differentiation]\label{lem:blockgrading}
The projectors and logarithms satisfy
\begin{equation}
 [P_i,\lambda]=0,\qquad
 [\mathscr L_i,\lambda]=zP_i,\qquad
 [G_a,P_i]=[G_a,\mathscr L_i]=0.
 \label{eq:block-commutators}
\end{equation}
In particular $P_i$ and $M_i=\mathscr L_i-zP_i\partial_\lambda$
are matrix series without coefficient differentiation. The
operator
\begin{equation}
 D_i=P_iG_a-\frac{\lambda}{z}\mathscr L_i
 \label{eq:block-grading}
\end{equation}
differentiates only in $z$, and $P_iD_i=D_iP_i=D_i$.
\end{proposition}

\begin{proof}
$[T_a,\lambda]=zT_a$. Apply \eqref{eq:functional-commutator} to
the locally-constant indicator and its supported logarithm.
$[G_a,T_a]=0$ and $[G_a,\xi]=0$ give $[G_a,P_i]=[G_a,\mathscr L_i]=0$.

At each coefficient of $P_i$ and $\mathscr L_i-zP_i\partial_\lambda$,
the identity $[D,\lambda]=0$ for finite-order $D$ forces $D$ to
have order zero (the leading coefficient of
$[a_J\partial_\lambda^J,\lambda]$ is $Ja_J\partial_\lambda^{J-1}$).
Writing out $G_a$ gives
\begin{equation}
 D_i=P_i z\partial_z+P_i(\tfrac12+\mu)
    +z^{-1}\bigl(P_i(c\star)-\lambda M_i\bigr).
 \label{eq:block-grading-matrix}
\end{equation}
No $\lambda$-derivatives remain, and neither $G_a$ nor the
functional calculus introduced derivatives in $y$ or $Q$. Support
$P_iD_i=D_iP_i=D_i$ follows from $[G_a,P_i]=0$,
$P_i\mathscr L_i=\mathscr L_iP_i=\mathscr L_i$, and
$[P_i,\lambda]=0$.
\end{proof}

\begin{proposition}[Modes and coefficient bounds]\label{lem:modes}
Define
\begin{equation}
 A_{-1,i}=z^{-1}P_i,\qquad
 A_{k,i}=z^{-1}(zD_i)^{k+1}\quad(k\geq 0).
 \label{eq:mode-definition}
\end{equation}
Each $A_{k,i}$ is a series in powers $z^m$, $m\geq -1$, with
coefficients that are matrix differential operators in
$z\partial_z$ of order at most $k+1$ and contain no derivatives
in $y,\lambda,Q$. Each fixed base and loop coefficient extends
holomorphically along paths in the generic locus of
\eqref{eq:spec-eigenvalues}, with the spectral labels and
logarithms continued along the path.

For any scalar $b$ independent of $z$, replacing $D_i$ by
$D_i+bz^{-1}P_i$ gives
\begin{equation}
 A_{k,i}\longmapsto\sum_{j=0}^{k+1}\binom{k+1}{j}
 b^{k+1-j}A_{j-1,i}.
 \label{eq:mode-binomial}
\end{equation}
For distinct branches $i\ne j$:
$(zD_i)(zD_j)=0$.
Consequently, if $P=\sum_iP_i$ and $D=\sum_iD_i$ over a group of
branches, the modes formed from $(P,D)$ are the sum of the
branches' modes.
\end{proposition}

\begin{proof}
Equation~\eqref{eq:block-grading-matrix} shows $zD_i$ has
nonnegative $z$-powers and Euler order at most one; products of
$z\partial_z$ preserve $z$-powers. Holomorphic continuation at
each fixed coefficient uses that the resolvent construction
involves finitely many inversions, parameter derivatives, and
contour residues; its only spectral denominators are the $a_i$,
which continue along any path on which they are distinct and
nonzero. This is a per-coefficient statement; no global
convergence asserted.

Since $D_i$ commutes with $b$ and $P_i$ is a two-sided identity
for $D_i$, the ordinary binomial formula applies in this supported
algebra, giving \eqref{eq:mode-binomial}. Finally,
$(zD_i)(zD_j)=zD_iP_izP_jD_j=0$ for $i\ne j$, and quantization
is linear in the modes.
\end{proof}

Changing a logarithm by $2\pi i n$ changes $D_i$ by
$-2\pi i n\lambda z^{-1}P_i$; the modes for one logarithm and any
other span the same collection. Infinitesimal symplecticity will
follow from the fixed-manifold comparison below; it is not needed
for the constructions here.

\subsection{Fixed-manifold blocks at \texorpdfstring{$y=0$}{y=0}}

Fix a germ of $\lambda\ne 0$. At $y=0$ one eigenvalue
$a_B=\lambda+h_B$ tends to zero and the remaining $r$ tend to
$\lambda$. Call the small branch the \emph{$B$ branch} and the
$r$-element group the \emph{$X$ cluster}. Choose disjoint contours
about $0$ and $\lambda$ enclosing these groups for $y$ small.
Both projectors extend through $y=0$; the logarithm applies
through $y=0$ on the $X$ contour (which avoids $0$). Denote the
resulting operators $P_B,P_X,\mathscr L_X$.

Switch the fixed-manifold coordinates (restriction to
$B\sqcup X$); the change is invertible at $\lambda\ne 0$ and
preserves the completions. Let $\Pi_B,\Pi_X$ be the constant
coordinate projectors and put
\[
 K_F(z,\lambda)=\prod_x
 \begin{cases}
  x+\lambda, & F=X,\\
  (x-\lambda-z)^{-1}, & F=B,
 \end{cases}
 \qquad E_z=e^{z\partial_\lambda},
\]
where $x$ ranges over the ordinary Chern roots of $N_F$. Then
\eqref{eq:shift-fixed} reads $T_d|_F=y^{j_F}K_FE_z$.

\begin{proposition}[Comparison at zero fiber degree]\label{prop:spec-zero}
The projectors $P_B,P_X$ and the cluster logarithm $\mathscr L_X$
have holomorphic coefficient germs through $y=0$. Their Taylor
expansions at $y=0$ satisfy
\begin{equation}
 P_F=S_W\Pi_FS_W^{-1}\ (F=B,X),\qquad
 \mathscr L_X=S_W\mathscr L_{d,X}S_W^{-1},
 \label{eq:spec-zero-conjugation}
\end{equation}
where $\mathscr L_{d,X}$ is the log of $K_XE_z$ on the $X$
summand, extended by zero on $B$.

$T_aP_B$ is divisible by $y$ coefficientwise in $z$ and base
degree. Set $\widetilde T_B=y^{-1}T_aP_B$. It has holomorphic
coefficient germs through $y=0$. Its nonzero block at
$y=z=Q^{>0}=0$ has eigenvalue $(-\lambda)^{-r}$ (possible
nilpotent part); let $\mathscr K_B$ be its supported logarithm.
Then on a punctured sector in $y$ one can choose the $B$-branch
logarithm as
\begin{equation}
 \mathscr L_B=(\log y)P_B+\mathscr K_B.
 \label{eq:spec-divided-log}
\end{equation}
The coefficients of $\mathscr K_B$ are holomorphic through $y=0$.
Its Taylor expansion, and hence \eqref{eq:spec-divided-log},
obeys the same conjugation by $S_W$ as in
\eqref{eq:spec-zero-conjugation}, with the fixed-block operators
in the descendant frame.
\end{proposition}

\begin{proof}
The reduction $T_a^{(0)}$ is a holomorphic matrix through
$y=0$; its spectrum there consists of $0$ and $\lambda$
(including multiplicities and nilpotent parts). The Neumann
construction with the two fixed contours gives the asserted
holomorphic extensions.

Taylor expanding coefficient germs in $y$ is a homomorphism
commuting with $\partial_\lambda$; it sends the ancestor
resolvent (nonnegative loop powers) to a Laurent series in the
full Novikov labels and preserves the inverse identities. Use
\eqref{eq:laurent} with $d=(\beta,l)$, $q^d=Q^\beta y^l$, and
coefficients holomorphic in the chosen $\lambda$-germ.
Rationality in Proposition~\ref{prop:loc-factorization} puts the
expansion of $S_W$ at $z=0$ in that algebra; its zero-curve
coefficient is the identity, so its inverse also sits in it. The
shift operators expanded via \eqref{eq:shift-fixed} also lie in
it, so their resolvents on the two contours lie in it. The
descendant resolvent uses the joint filtration by $z$ and full
Novikov degree: its initial $B$ block is zero and its initial $X$
block is $\lambda+p$, whose nilpotent part is the ordinary class
$p|_X$.

\eqref{eq:shift-calibration} and Lemma~\ref{lem:gauge} now
identify the actual resolvents in the two frames. In the
descendant frame the contour about zero selects the whole $B$
summand, and the contour about $\lambda$ selects the whole $X$
summand (within each summand the contour function is identically
$1$ or $0$, so Lemma~\ref{lem:calculus} gives identity or zero
also for the formal perturbation). Integration gives
\eqref{eq:spec-zero-conjugation}, including logarithms on $X$.

On the $B$ summand $T_d\Pi_B=yK_BE_z\Pi_B$; $y$ is central. So
$T_aP_B=yS_WK_BE_z\Pi_BS_W^{-1}$ in the Laurent algebra completed
in full Novikov degree. Neither $S_W$ nor its inverse has negative
$y$-powers, so the constant Taylor coefficient in $y$ of every
base and loop coefficient on the left vanishes. Those
coefficients are already holomorphic through $y=0$, so division
gives $\widetilde T_B=S_WK_BE_z\Pi_BS_W^{-1}$ (after Taylor
expansion).

At $y=z=Q^{>0}=0$, the calibration is the identity and the
nonzero block of the previous identity is $\prod_x(x-\lambda)^{-1}$.
Ordinary positive-degree classes are nilpotent, so the sole
spectral value is $(-\lambda)^{-r}\ne 0$. Choose a contour
about this value avoiding zero and a logarithm there;
Lemma~\ref{lem:calculus} defines $\mathscr K_B$ holomorphically
through $y=0$ with the projector $P_B$ as its two-sided identity
(by Proposition~\ref{prop:spec-zero} and Lemma~\ref{lem:gauge}).

Finally, restrict to a punctured sector and choose $\log y$. On
the $P_B$-supported algebra $T_a=y\widetilde T_B$. Rescaling
the spectral contour by the central scalar $y$ and choosing
$\log(y\xi)=\log y+\log\xi$ shows that its logarithm is exactly
\eqref{eq:spec-divided-log}. Conjugation follows because
multiplication by $\log y$ commutes with everything.
\end{proof}

Near $y=0$ the $B$-branch modes are finite polynomials in
$\log y$ with holomorphic coefficients: write
$D_B=D_B^{\mathrm{reg}}-\lambda(\log y)z^{-1}P_B$ with
$D_B^{\mathrm{reg}}=P_BG_a-(\lambda/z)\mathscr K_B$ holomorphic,
then apply \eqref{eq:mode-binomial} with $b=-\lambda\log y$. For
$A_{k,B}$ the logarithmic degree is at most $k+1$. The $X$-cluster
modes are holomorphic through zero. None of this requires
convergence in $z$ or an analytic meaning for the Novikov series.

% ==================================================================
\section{The ordinary theory on a fixed component}\label{sec:fixed}
% ==================================================================

The operators of Section~\ref{sec:spec} were built from the
equivariant theory of $W$. This section identifies what their
constraints mean near $y=0$: on a fixed component, Quantum
Riemann--Roch removes the inverse-Euler twist, and the logarithm
of the shift becomes a translation generator in $\lambda$. Its
derivative term cancels the derivative term of the equivariant
grading, leaving the ordinary grading plus a scalar multiple of
$z^{-1}$.

\begin{proposition}\label{prop:fixed-equivalence}
Fix a germ at $\lambda\neq 0$. Near $Q=y=0$, use the full Novikov
completion with a formal $\log y$ adjoined, allowing polynomial
dependence on $\log y$ in each coefficient. Let $F=B$ denote the
branch near the eigenvalue zero, or $F=X$ the whole cluster near
the eigenvalue $\lambda$. Use the projectors and logarithms of
Proposition~\ref{prop:spec-zero}, and let $A_{k,F}$ be their modes.
Then, over this coefficient ring,
\[
 \mathcal A_W \text{ satisfies every }A_{k,F}\text{ projectively}
 \;\Longleftrightarrow\;
 Z_F \text{ satisfies every }\ell_{k,F}\text{ projectively},
 \quad k\geq -1.
\]
The operators on the left are infinitesimal symplectic for the
equivariant pairing of $W$. On the right, $Z_F$ is the ordinary
descendant potential with its full cohomology and actual integral
curve labels.
\end{proposition}

The proof has two parts. First we show that the fixed-block
operators, after a change of paired space, become the ordinary
Virasoro modes plus a scalar shift. Then we show that this change
of space is realized by the quantum Riemann--Roch multiplier and,
after separating the rank factor, that the resulting quantized
equations are those of the actual Euler twist.

\subsection{Removing the normal bundle from the grading}

Fix $F$ and abbreviate $n=n_F$, $w=w_F$, $j=j_F$. Put
\[
 a=w\lambda,\qquad c_N=c_1(N_F)_{\mathrm{ord}},\qquad W_F=nw.
\]
After restriction to the descendant space of $F$, the grading and
shift from Section~\ref{sec:shift} are
\begin{align}
 G_F&=z\partial_z+\lambda\partial_\lambda+\tfrac12+\mu_F-\tfrac n2
      +\frac{R_F+c_N+nw\lambda}{z},
 \label{eq:restricted-grading}\\
 T_F&=y^j\prod_x
     \begin{cases}
      x+\lambda, & w=1,\\
      (x-\lambda-z)^{-1}, & w=-1
     \end{cases}
     e^{z\partial_\lambda}.
 \label{eq:restricted-shift}
\end{align}
Here $x$ runs over the ordinary Chern roots of $N_F$; symmetric
expressions are read via the splitting principle, and all
$x$-expansions are finite on $H^*(F)$.

Choose a germ of $\log a$ at a nonzero $\lambda$. Define a
multiplier $\Delta_F$ by
\begin{equation}\label{eq:Delta-F}
 \log\Delta_F=\sum_x\Bigl\{\tfrac12\log(a+x)
 +\frac1z\int_0^x-\log(a+t)\,dt
 +\sum_{m\geq 1}\frac{B_{2m}}{(2m)!}z^{2m-1}
   \partial_x^{2m-1}\bigl(-\log(a+x)\bigr)\Bigr\},
\end{equation}
with $B_{2m}$ the Bernoulli numbers. The $z^{-1}$-coefficient
has positive ordinary cohomological degree, hence is a nilpotent
multiplication; both $\Delta_F$ and $\Delta_F^{-1}$ have a finite
$z$-lower bound, and positive tails are formal power series at
$z=0$.

The multiplier maps the ordinary paired loop space of $F$ to the
space with pairing $\eta_F^t$. Indeed, all terms of
\eqref{eq:Delta-F} except the half-logarithm are odd in $z$, so
\[
 \Delta_F(-z)\Delta_F(z)=\prod_x(a+x)
 =e_T(N_F),
\]
and the inverse Euler factor in $\eta_F^t$ cancels this product.
This fixes the direction of the square-root factor.

For the scalar part of the shift, put
\begin{equation}\label{eq:Theta-F}
 L(v)=v-v\log v,\qquad \Theta_F(\lambda)=nL(w\lambda).
\end{equation}

\begin{lemma}\label{lem:fixed-classical}
The following hold in the Laurent calculus of
Lemma~\ref{lem:gauge}:
\begin{align}
 \Delta_F^{-1}G_F\Delta_F
 &=\ell_{0,F}+\lambda\partial_\lambda+\frac{nw\lambda}{z},
 \label{eq:fixed-grading-conjugation}\\
 \Delta_F^{-1}T_F\Delta_F
 &=y^j\exp\Bigl(\frac{\Theta_F(\lambda)
      -\Theta_F(\lambda+z)}{z}\Bigr)
   e^{z\partial_\lambda}.
 \label{eq:fixed-shift-conjugation}
\end{align}
Under the same change of space, the logarithm prescribed near
$y=0$ becomes
\begin{equation}\label{eq:fixed-log-conjugation}
 z\partial_\lambda+j\log y-\Theta_F'(\lambda)
 =z\partial_\lambda+j\log y+nw\log(w\lambda),
\end{equation}
up to a scalar constant set by the logarithm branch.
\end{lemma}

\begin{proof}
The cohomological commutator with $\mu_F$ differentiates functions
of a root by $x\partial_x$ because $x$ has Hodge type $(1,1)$.
Apply $z\partial_z+\lambda\partial_\lambda+x\partial_x$ to a
summand of \eqref{eq:Delta-F}. The half-logarithm contributes
$1/2$. For $I(a,x)=\int_0^x-\log(a+t)dt$ we have
$(\lambda\partial_\lambda+x\partial_x)I=I-x$; its contribution
is $-x/z$. Each Bernoulli term has total degree zero. Summing
over roots gives $n/2-c_N/z$, canceling the two corresponding
terms in \eqref{eq:restricted-grading}, giving
\eqref{eq:fixed-grading-conjugation}.

For the shift, add $L(a)/z$ to the summand indexed by $x$ in
\eqref{eq:Delta-F}. The sum depends only on $v=a+x$ and equals
\[
 \Phi(v)=\Bigl((z\partial_v)^{-1}-\tfrac12+\sum_{m\geq 1}
  \frac{B_{2m}}{(2m)!}(z\partial_v)^{2m-1}\Bigr)(-\log v),
\]
where the antiderivative is $L(v)/z$. The Bernoulli generating
series gives $\Phi(v+z)-\Phi(v)=-\log v$. When $w=1$, this
difference cancels the factor $v$ in \eqref{eq:restricted-shift};
when $w=-1$, use $\Phi(v-z)-\Phi(v)=\log(v-z)$ to cancel the
factor $(v-z)^{-1}$. The added terms $L(a)/z$ account for the
exponential in \eqref{eq:fixed-shift-conjugation}.

For the logarithm (as opposed to the exponential form), set
$H=z\partial_\lambda-\Theta_F'(\lambda)$. Its evolution is
\begin{equation}\label{eq:fixed-log-flow}
 e^{tH}=\exp\Bigl(\frac{\Theta_F(\lambda)-\Theta_F(\lambda+tz)}
 {z}\Bigr)e^{tz\partial_\lambda};
\end{equation}
differentiating the right side in $t$ gives $H$ times that side,
and the value at $t=0$ is the identity. The functional calculus
of Lemma~\ref{lem:calculus} agrees (coefficientwise
differentiation of the contour integral gives the same evolution).
The $z$-constant term is $-\Theta_F'(\lambda)$; locally choose a
logarithm that inverts the exponential at that scalar; the
composition rule in Lemma~\ref{lem:calculus} then identifies
$\log(e^H)$ with $H$. Conjugating by $\Delta_F$ is allowed by
Lemma~\ref{lem:gauge}. For $F=B$, apply this to the divided shift
$y^{-1}T_F$ and restore $\log y$; for $F=X$, apply it directly to
$T_F$. These are exactly the logarithm prescriptions of
Proposition~\ref{prop:spec-zero}, giving
\eqref{eq:fixed-log-conjugation} up to the logarithm-branch
scalar.
\end{proof}

Subtracting $\lambda/z$ times \eqref{eq:fixed-log-conjugation}
from \eqref{eq:fixed-grading-conjugation} removes
$\lambda\partial_\lambda$. Thus the operator $D_F$, after the
descendant change of space and then $\Delta_F$, becomes
\begin{equation}\label{eq:fixed-ordinary-grading}
 \ell_{0,F}+\frac{b_F}{z},\qquad
 b_F=\lambda\bigl(nw-j\log y-nw\log(w\lambda)\bigr),
\end{equation}
up to a scalar multiple of $\lambda/z$ from a logarithm choice.
In particular it differentiates neither $\lambda$, $y$, nor the
Novikov variables.

For a scalar $b$ independent of $z$, write
\[
 \ell_{-1,F}^{(b)}=z^{-1},\qquad
 \ell_{k,F}^{(b)}=z^{-1}(z\ell_{0,F}+b)^{k+1}\quad(k\geq 0).
\]
The binomial identity
\begin{equation}\label{eq:fixed-binomial}
 \ell_{k,F}^{(b)}=\sum_{m=-1}^{k}\binom{k+1}{m+1}
 b^{k-m}\ell_{m,F}
\end{equation}
is triangular with diagonal entries one. Consequently these
operators are infinitesimal symplectic, and their simultaneous
projective equations are equivalent to those of the ordinary
modes. This completes the classical identification. Now we lift
it to the potentials.

\subsection{Quantizing the comparison}

The $\Delta_F$ above contains $-\log(w\lambda)c_N/z$, a
classical multiplication that we cannot directly exponentiate in
$\lambda^{-1}$. We therefore first use the normalized class
\begin{equation}\label{eq:normalized-class}
 c_0(N_F)=\prod_x(1+x/a)^{-1},
\end{equation}
with $Z_{F,\mathrm{ntw}}$ and $\eta_F^{\mathrm{ntw}}$ the
corresponding descendant potential and pairing. Define
$\Delta_{F,0}$ by \eqref{eq:Delta-F} with $-\log(a+x)$ replaced by
$-\log(1+x/a)$ and $\tfrac12\log(a+x)$ by $\tfrac12\log(1+x/a)$.
It maps the ordinary paired space to
$(\cH_F,\eta_F^{\mathrm{ntw}})$ and is the identity modulo
$\lambda^{-1}$.

\begin{lemma}[Quantum Riemann--Roch in the normalized twist]\label{lem:qrr}
With quantization and dilaton translations as in
Section~\ref{sec:conv},
\begin{equation}\label{eq:qrr}
 Z_{F,\mathrm{ntw}}=(\text{invertible scalar})\,
  U_{\Delta_{F,0}}Z_F.
\end{equation}
This is an invertible identity formal in $\lambda^{-1}$ and
coefficientwise in the full Novikov variables and $\hbar$.
\end{lemma}

\begin{proof}
Quantum Riemann--Roch of Coates--Givental says that for a
multiplicative class whose logarithm on a line with first Chern
class $x$ is $s(x)$, its multiplier (in the fixed ordinary
pairing) has exponent
\[
 \sum_x\Bigl\{\frac1z\int_0^x s(t)dt
   +\sum_{m\geq 1}\frac{B_{2m}}{(2m)!}z^{2m-1}s^{(2m-1)}(x)\Bigr\},
\]
and it identifies the twisted and ordinary pairings by
multiplication by $\sqrt{c_0(N_F)}$. Expressing the conclusion
as a map \emph{to} the actual twisted space adds $-s(x)/2$ to the
exponent. Taking $s(x)=-\log(1+x/a)$ gives exactly
$\Delta_{F,0}$.

The affine-coordinate bookkeeping matches: in the ordinary
Fock space the twisted potential uses
$\widetilde q=\sqrt{c_0(N_F)}(t-z1_F)$, and multiplication by
$c_0(N_F)^{-1/2}$ returns the twisted Darboux coordinate
$q_{\mathrm{ntw}}=t-z1_F$. Thus $U_{\Delta_{F,0}}$ has the
dilaton translations of \eqref{eq:qrr}. The theorem applies to
$R\pi_*\ev^*N_F$ on the ordinary stable-map spaces of the smooth
projective $F$, is invertible and formal in $\lambda^{-1}$, and
uses the categorical inverse for nodal contractions. So it holds
in our Koszul conventions, for arbitrary Hodge types and parity.
\end{proof}

The two classical multipliers are related simply:
\begin{equation}\label{eq:Delta-normalization}
 \Delta_F=a^{n/2}\exp(-\log(a)c_N/z)\,\Delta_{F,0}.
\end{equation}
If $C$ is an ordinary divisor class, then
$[\ell_{0,F},C/z]=[z\partial_z,C/z]+[\mu_F,C/z]=0$ and
$[z,C/z]=0$ (the two contributions being $-C/z$ and $+C/z$,
since $R_F$ commutes with $C$). So $C/z$ commutes with all
ordinary modes and with their scalar translates
\eqref{eq:fixed-binomial}. After the coefficient derivatives are
removed in \eqref{eq:fixed-ordinary-grading}, the scalar $a^{n/2}$
also cancels in conjugation. Consequently $\Delta_F$ and
$\Delta_{F,0}$ produce the same conjugated mode matrices. By
Lemma~\ref{lem:qrr} and Lemma~\ref{lem:covariance}, the equations
for those matrices on $Z_{F,\mathrm{ntw}}$ are equivalent to the
ordinary projective equations on $Z_F$.

It remains to restore the rank factor omitted in
\eqref{eq:normalized-class}. This changes the paired space as
well as the invariants, and both must be handled together.

\begin{lemma}[Rank scaling]\label{lem:rank-scaling}
Put $m_F=a^{-n}$. The actual inverse-Euler twist is obtained from
the normalized twist by the substitutions
\begin{equation}\label{eq:scaling}
 \hbar\longmapsto\hbar/m_F,\qquad
 Q^\beta y^l\longmapsto Q^\beta y^l a^{-\int_d c_N}
\end{equation}
(where $d$ is the corresponding actual class on $F$). Its pairing
is $\eta_F^t=m_F\eta_F^{\mathrm{ntw}}$. Under these substitutions
the quadratic quantizations of
$\Delta_{F,0}\ell_{k,F}\Delta_{F,0}^{-1}$ (for every $k\geq -1$)
in the two paired spaces agree.
\end{lemma}

\begin{proof}
On $\overline{\mathcal M}_{g,m}(F,d)$, Riemann--Roch gives
$\rank R\pi_*\ev^*N_F=n(1-g)+\int_d c_N$. The ratio of the
inverse-Euler class to the normalized class is therefore
$a^{-n(1-g)-\int_d c_N}=m_F^{1-g}a^{-\int_d c_N}$, exactly the
change in the coefficient of $\hbar^{g-1}Q^\beta y^l$ under
\eqref{eq:scaling}.

The $qq$ part of quantization uses $\eta/\hbar$; the $pp$ part
uses $\hbar\eta^{-1}$. In the actual twist these become
$m_F\eta^{\mathrm{ntw}}/\hbar$ and
$(\hbar/m_F)(\eta^{\mathrm{ntw}})^{-1}$, exactly the normalized
counterparts at $\hbar/m_F$. The mixed part is unchanged. Since
$\Delta_{F,0}\ell_{k,F}\Delta_{F,0}^{-1}$ has no Novikov
derivatives or Novikov dependence, the curve substitution
commutes with its action. Restore the scalar $b_F$ afterwards by
\eqref{eq:fixed-binomial}. Each curve monomial is multiplied by a
nonzero scalar, so the substitution is injective and reversible
without identifying curve classes.
\end{proof}

\paragraph{Rationality in $\lambda$.} The above used QRR only as a
formal identity in $\lambda^{-1}$. To pass to germs at nonzero
$\lambda$, first remove $b_F$ by \eqref{eq:fixed-binomial}. Each
coefficient of the conjugated matrices
$\Delta_{F,0}\ell_{k,F}\Delta_{F,0}^{-1}$ is then rational in
$\lambda$; the matrices have finite $z$-lower bounds. The
coefficients of $Z_{F,\mathrm{tw}}$ are also rational in
$\lambda$ (the torus acts trivially on $F$, the inverse Euler
class expands only up to finite cohomological dimension, and
descendant indices are bounded at fixed genus, class and number of
marks). Then each nonconstant coefficient of
$Z_{F,\mathrm{tw}}^{-1}\op_{\eta_F^t}
(\Delta_{F,0}\ell_{k,F}\Delta_{F,0}^{-1})Z_{F,\mathrm{tw}}$
is a finite sum of rational functions of $\lambda$; vanishing of
its Laurent expansion at infinity is equivalent to vanishing as a
germ. At each genus and curve coefficient, \eqref{eq:scaling}
only shifts finitely many $\lambda$-powers, so the argument runs
in both directions. We then restore $b_F$ and its logarithms via
\eqref{eq:fixed-binomial}. No analytic value of the quantized
infinite multiplier is taken.

\subsection{Proof of Proposition~\ref{prop:fixed-equivalence}}

Let $F'$ be the other fixed component. Up to invertible scalars,
the potential chain is
\[
\begin{gathered}
 Z_F\xrightarrow{U_{\Delta_{F,0}}}Z_{F,\mathrm{ntw}}
    \xrightarrow{\eqref{eq:scaling}}Z_{F,\mathrm{tw}},\\
 Z_{F,\mathrm{tw}}\xrightarrow{U_{S_f}}
    \mathcal A_{F,\mathrm{tw}}(u_F),\\
 \mathcal A_{F,\mathrm{tw}}(u_F)\,\mathcal A_{F',\mathrm{tw}}(u_{F'})
    \xrightarrow{U_R}\mathcal A_W.
\end{gathered}
\]
The arrow \eqref{eq:scaling} changes parameters and rescales the
pairing per Lemma~\ref{lem:rank-scaling}. The final row first
adjoins the other fixed potential and then applies the upper
symplectic transformation.

\eqref{eq:fixed-binomial}, Lemma~\ref{lem:qrr} and
Lemma~\ref{lem:rank-scaling} identify projective satisfaction of
all ordinary modes on $Z_F$ with projective satisfaction of the
corresponding descendant modes on $Z_{F,\mathrm{tw}}$. The
rationality argument above gives the equivalence over the same
germs in $\lambda$ used by the spectral construction.

Next, pass from descendants on $F$ to ancestors at $u_F$: the
ancestor--descendant identity \eqref{eq:ancestor-descendant} and
Lemma~\ref{lem:covariance} conjugate the mode matrices by
$S_f=S_{F,\mathrm{tw}}(u_F,z)$ and transport their projective
equations to $\mathcal A_{F,\mathrm{tw}}(u_F)$. On the product of
the two fixed potentials, an operator supported on $F$ acts only
on that factor, so its projective equation is equivalent to the
equation on the single factor.

Finally, apply $R(z)$ of Proposition~\ref{prop:loc-factorization}.
The identities
\[
 S_W=R(z)\bigoplus_F S_f,\qquad
 \mathcal A_W=(\text{scalar})\,U_R\prod_F\mathcal A_{F,\mathrm{tw}}(u_F)
\]
show, by Proposition~\ref{prop:spec-zero}, that the resulting
classical modes are exactly $A_{k,F}$ and their quantum equations
are the projective equations on $\mathcal A_W$. Each change is
invertible, so the implication holds both ways.

All quantized changes are defined in the Novikov completion:
$u_F$, $S_f-I$, and $R-I$ have positive Novikov filtration; $S_f$
has a finite negative tail at each curve coefficient; $R$ is
upper. Conjugation and quantization preserve the lower loop
bounds and finite support required in
Section~\ref{sec:conv}. The comparison uses the upper
quantization from localization plus the fixed-theory ancestor
changes; it does not require a new expansion of $S_W$.
$\square$