As $v\subset \C^*\cdot p$, it must lie in the same connected component of torsion points as $p$.
\end{proof}

\subsection{Floer solutions ``consume filtration'' when crossing linearity regions}
\label{Subsection Floer solutions consume filtration when crossing linearity regions}

So far we have not needed to specify the choices of constants $\phi_i$ and $\varepsilon_i$ in the construction of $\phi$ and $L_i$. We now show that one can choose the $\phi_i$ inductively so that Floer solutions must consume an a priori lower bound of filtration-energy when crossing the ``linearity'' region above $L_i$ (so when $R\circ \Psi\in L_i$). The reason we wish to have this result, is that if we modify $H_{\lambda}$ at infinity by increasing the slope to obtain $H_{\lambda'}$, we would like $CF^*(H_{\lambda})$ to be a subcomplex of $CF^*(H_{\lambda'})$. For convex symplectic manifolds this essentially follows from the maximum principle for radial Hamiltonians, but in our setup the maximum principle only holds over the linearity regions, so we instead resort to a filtration argument. This argument will also be needed to ensure that certain spectral sequences are compatible with increasing the slope of the Hamiltonian, so that one can take a direct limit of the spectral sequences.

We refine the choices of $\varepsilon_i$, $N_i$, so that now $\phi$ belongs to the class of cut-off functions satisfying:
\begin{enumerate}
\item[(4)] $\phi\geq R$ for $R \geq R_1'$.
\item[(5)] $\phi'\geq 1$ on a subinterval of $L_i$ of width at least $\varepsilon_i/2$.
\end{enumerate}
This can be achieved by choosing $N_0,\ldots,N_i$ to be small, $\varepsilon_i=\textrm{width}(L_i)$ to be large, and by making $\phi'$ very large on a large subinterval of $L_i$, in particular making $\phi_i$ large enough so that the constant value of $\phi$ on $N_{i+1}$ is larger than $R_{i+1}''=\max_{N_{i+1}} R$.

\begin{lm}\label{LemmaDeltaiFiltration}
Given $\delta_i>0$, and any choice of $\phi$ defined on $R\leq R_i''$ within the above class, there is a way to extend $\phi$ within that class over $L_i=[R_i'',R_{i+1}']$ so that any Floer trajectory $u:\R\times S^1 \to Y$ from $x_-$ to $x_+$ for $H_{\lambda}$ which crosses the region above $L_i$ satisfies $F(x_-)-F(x_+)>\delta_i$.

More precisely, if there are values $s_0<s_1\in \R \cup \{\pm \infty\}$ for which the loops $u(s_0,\cdot)$ and $u(s_1,\cdot)$ project via $R\circ \Psi$ to different components of $\R \setminus L_i$,
\begin{equation}\label{Equation filtration jump of Floer when crossing region}
F(x_-)-F(x_+)\geq F(u(s_0,\cdot))-F(u(s_1,\cdot)) > \delta_i.
\end{equation}
\end{lm}
\begin{proof}
As \eqref{nonpositivityOfOmegaV} is pointwise non-negative, it suffices that we bound the integral
$\int \eta(\partial_s v, \partial_t v - \lambda_i \mathcal{R}_B)$
from below by $\delta_i$ over the subset of all $(s,t)\in \R \times S^1$ for which $R(v(s,t))$ lies in the subinterval of $L_i$ where we ensured that $\phi'\geq 1$ (and $h'=\lambda_i$). For such $(s,t)$, using
$\phi\geq R$ on $R\geq R_1'$,
\eta(\partial_s v, \partial_t v - \lambda_i \mathcal{R}_B)
\geq R \cdot d\alpha(\partial_s v,\J_B\partial_s v) + (dR \wedge \alpha)(\partial_s v,\J_B\partial_s v) = d(R\alpha)(\partial_s v, \partial_t v - \lambda_i \mathcal{R}_B).
The latter expression is precisely the integrand of the energy $\int \|\partial_s v\|^2 \, ds\, dt$ for a Floer solution $v$ in $B$ for the Hamiltonian $\lambda_i R$. So the problem reduces to showing that a Floer solution in $B$ for a radial Hamiltonian consumes a lot of energy if it crosses a long radial stretch in which the slope of the Hamiltonian is constant, as we can choose the width $\varepsilon_i=\textrm{width}(L_i)$ to be arbitrarily large. This is a standard ``monotonicity lemma argument'': one uses Gromov's trick of turning a Floer solution in a symplectic manifold $M$ into a {\ph } section of a bundle with fibre $M$ (e.g.\,see \cite[Sec.5.3]{R14}). One then applies the monotonicity lemma to obtain lower bounds on the energy (see  \cite[Lem.33]{R14}). This is the step that uses the assumption in \cref{Def:KahlerMfdWithProjection} that $(B,I_B)$ is geometrically bounded at infinity, and the observation that in the auxiliary almost complex structure in Gromov's trick \cite[Def.26]{R14} the off-diagonal terms involve $\lambda_i \mathcal{R}_B$ and $\lambda_i Z_B$ which are radially invariant.
\end{proof}

\begin{rmk}\label{Remark montone hpy is ok}
Lemma \ref{LemmaDeltaiFiltration} also holds for Floer continuation solutions for homotopies $H_s$ as in Lemma \ref{Lemma Floer continuation maps are ok}. In the proof, on the interval $L_i$ we have $h_s' = \lambda_{i,s}$, and the condition $\partial_s \lambda_{i,s}\leq 0$ ensures that the monotonicity argument still applies
(see the discussion of monotone homotopies in \cite[Sec.5.3]{R14}).
\end{rmk}

\subsection{Cofinal family of Hamiltonians $H_{\lambda_i}$, and a filtration on $SH^*$}
\label{Subsection cofinal Hams}

To compute $SH^*(Y,\varphi)$ it suffices to take a direct limit over a cofinal family of Hamiltonians $H_{\lambda_i}$, meaning $\lambda_i \to \infty$ as $i \to \infty$. Such $H_{\lambda_i}$ can be constructed so that the Floer continuation maps are particularly well-behaved.

\begin{cor}\label{Cor Subcomplex Trick}\label{CorSubcomplexTrick}
There is a cofinal sequence of Hamiltonians $H_{\lambda_i}$ and a cut-off function $\phi$, such that $CF^*(H_{\lambda_i})\subset CF^*(H_{\lambda_{i+1}})$ is a subcomplex. There are filtration-preserving Floer continuation maps $CF^*(H_{\lambda_i})\to CF^*(H_{\lambda_{j}})$ for $i<j$ which equal the inclusion maps of subcomplexes.
The union $$SC^*(Y,\varphi):=\cup\, CF^*(H_{\lambda_i})$$ of these nested complexes is a filtered complex, and $SH^*(Y,\varphi)$ is (isomorphic to) its cohomology.
Analogously $SC^*_+(Y,\varphi):=\cup\, CF^*_+(H_{\lambda_i})$ is a filtered complex with cohomology $SH^*_+(Y,\varphi)$.
\end{cor}
\begin{proof}
We build $H_{\lambda_i}$ inductively as in \cref{SectionConstructionOfHLambda} so that $H_{\lambda_i}=\lambda_i H + \mathrm{constant}$ for $H\geq H_i''$, and $H_{\lambda_{i+1}}$ is obtained from $H_{\lambda_i}$ by increasing its (generic) slope at infinity to $\lambda_{i+1}>\lambda_i$.
Inductively, we ensure that $H_{\lambda_{i+1}}=H_{\lambda_i}$ for $H\leq H_i''$. The construction of $H_{\lambda_{i+1}}$ involves a choice of new parameters $H_{i+1}',H_{i+1}'',R_{i+1}',R_{i+1}''$ in \cref{SectionConstructionOfHLambda}, in particular a choice of width $\varepsilon_i=\mathrm{width}(L_i)$.
We build the Floer continuation map by considering a homotopy $H_s$ from $H_{\lambda_{i+1}}$ to $H_{\lambda_i}$ that decreases the slope of $H_{\lambda_{i+1}}$ back down to $\lambda_i$ on the region $H \geq H_i''$, in particular over $L_i$ we ensure that  $H_s=\lambda_s H +\mathrm{constant}(s)$ with $\partial_s \lambda_s \leq 0$.
The continuation map then respects the filtration, by Lemma \ref{Lemma Floer continuation maps are ok}.
If the continuation map were not the inclusion, there would exist a Floer continuation solution $u$ from a $1$-orbit $x_-\in CF^*(H_{\lambda_{i+1}})$ with $S^1$-period value $T_{-(i+1)}$, to a $1$-orbit
$x_+\in CF^*(H_{\lambda_i})$; we need to prevent that.
Such a Floer solution $u$ crosses the linearity region $L_i$ in the construction of $H_{\lambda_{i+1}}$. By \cref{Remark montone hpy is ok}, picking $\delta_i$ to be larger than the possible values $F(x_-)-F(x_+)$ for $1$-orbits $x_-$ of $H_{\lambda_{i+1}}$, with $S^1$-period value $T_{-(i+1)}$, and $1$-orbits $x_+$ for $H_{\lambda_i}$, a Floer continuation solution that crosses the region over $L_i$ would contradict  inequality \eqref{Equation filtration jump of Floer when crossing region}.
So such Floer continuation solutions are trapped in the region $H\leq H_i''$ where $H_s$ is $s$-independent. But such Floer  continuation solutions are constant as they cannot be isolated, due to the $\R$-reparameterisation in $s$. Thus $u$ does not cross $L_i$ after all. %
\end{proof}

\begin{rmk}
The previous proof cannot simply be achieved by a maximum principle, since the Floer continuation solution may achieve a maximum $(R\circ \Psi)$-value in the region where $c'$ is not locally constant, in which case the maximum principle for the $\Psi$-projection to $B$ is not known.
\end{rmk}

\section{Transversality for Floer solutions}
\label{Section Transversality for Floer solutions}

\subsection{Overview of the technical difficulty}

As mentioned in \cref{Rmk technical symplectic assumptions on Y}, transversality for Floer solutions can be achieved by the standard methods of [16, 44] for $Y$, by $C^{\infty}$-small perturbations $I+\varepsilon$ of $I$ (where $\varepsilon$ is an operator of small norm, such that $I+\varepsilon$ is an $\omega$-compatible almost complex structure). However, such a perturbation will in general ruin the holomorphicity of the map $\Psi$, and the projection $v=\Psi\circ u$ of a Floer solution $u\subset Y$ would only satisfy an equation in $B$ of type
$$\partial_t v - k(s,t)X_{S^1,B} = I_B\partial_s v+w(s,t)$$
where $w(s,t)=\Psi_*\circ \varepsilon_{u(s,t)} \circ \partial_s u$.
This may ruin the estimate in \cref{nonpositivityOfOmegaV} due to the term $d\alpha(\partial_s v, w(s,t))$. Even in a region where $\phi'\neq 0$, so where $\eta$ is symplectic, the term $\eta(\partial_s v, I_B\partial_s v)\geq 0$ may be too small to off-set a negative term $d\alpha(\partial_s v, w(s,t))$. Specifically, a large $\|\partial_s u\|$ has small $\|\partial_s v\|$ when $\partial_s u$ lies close to $\ker \Psi_*$, so $\|w\|$ cannot a priori be bounded by $\|\partial_s v\|$ by making $\|\varepsilon\|$ small, unless $\varepsilon$ preserves $\ker \Psi_*$. Once we demand that $\varepsilon$ preserves $\ker \Psi_*$ we run into two problems. Firstly, $\ker \Psi_*$ is not a locally trivial fibration, e.g.\,$\dim \ker \Psi_*|_p$ is only known to be upper semi-continuous in $p\in Y$, so it is not always possible to extend a vector $v\in \ker \Psi_*|_p$ to a local section of $\ker \Psi_*$. Secondly, the standard construction of the perturbation of $I$ in \cite[pp.1345-1346]{Salamon-Zehnder} and \cite[p.268]{FloerHoferSalamon} seems to run into problems when forcing $\varepsilon$ to preserve $\ker \Psi_*$, for instance if the adjoint $L^2$-section along $u$ in that argument were entirely contained in $\ker \Psi_*\subset TY$.

We will resolve these issues in an indirect way, by using a Gromov compactness argument combined with the use of a construction similar to the filtered Floer homology introduced by Ono \cite[Sec.3]{Ono}.
Consider what happens to a sequence of Floer solutions $u=u_{\varepsilon}$ of $(H_{\lambda},I+\varepsilon)$ for a sequence of perturbations $\varepsilon\to 0$ converging to zero. Let us assume for simplicity that the Hamiltonian $1$-orbits $x_{\pm}$ at the ends of $u_{\varepsilon}$ are fixed.
If we knew that the $u_{\varepsilon}$ converged, in the sense of Gromov compactness, to a broken Floer solution for $(H_{\lambda},I)$, then we would deduce that the filtration satisfies the correct inequality $F(x_-)\geq F(x_+)$ by \cref{H_lambdaIsOneDirected}.
Gromov compactness applies to Floer solutions with bounded energy
so, as explained in \cref{Subsection Gromov compactness when the energy is bounded},
one achieves the correct filtration inequality for all Floer solutions $u$ of $(H_{\lambda},I+\varepsilon)$ of energy $E(u)\leq E_0$
for any given perturbation $\varepsilon$ that is sufficiently small. However, how small $\varepsilon$ needs to be will a priori depend on the choice of energy bound $E_0$, so it does not rule out the possibility of a sequence of Floer solutions $u_{E_0}$ of energy $E(u_{E_0})>E_0$ that defy the filtration inequality, and which do not have a subsequence with a meaningful Gromov limit.
We construct a filtered version of the Floer chain complex, which artificially creates the necessary energy bounds so that Floer cohomology is obtained by a limiting procedure from Floer cohomology groups that only count Floer trajectories below a certain energy bound.

\subsection{Gromov compactness when the energy is bounded}\label{Subsection Gromov compactness when the energy is bounded}

From now on, we continue with the assumptions on $Y$ from the start of \cref{filtrationFloer}.
By the maximum principle from [41], Floer solutions for $(H_{\lambda}, I)$ remain in a compact region $K\subset Y$, so regularity of Floer solutions can be achieved by perturbing $I$ in $K$. We want to do this perturbation compatibly with the filtration:

\begin{lm}\label{Lemma perturb I subject to energy bound}
Given an admissible Hamiltonian $H_\lambda$, a
constant $E_0>0$, and a compact region $K\subset Y$, any sufficiently $C^{\infty}$-small perturbation $I'$ supported on $K$ of the $\omega$-compatible almost complex structure $I$ satisfies the filtration inequality \eqref{FiltrationByF}, $F(x_-)\geq F(x_+)$, for all Floer trajectories $u$ of $(H_{\lambda},I')$ of energy $E(u)\leq E_0$.

If $K$ contains all $1$-periodic orbits of $H_{\lambda}$ in its interior, then any generic perturbation $I'$ will also satisfy transversality for all Floer solutions.

Moreover, there is a $\delta_E>0$ such that for any $s$-dependent perturbation $I_s'$ (compactly supported in $s$) homotopying between two given perturbations $I'_-,I'_+$, with $\|I_s - I\|<\delta_E$, all Floer continuation solutions for $(H_{\lambda},I_s)$ of energy at most $E$ will satisfy the filtration inequality.
\end{lm}
\begin{proof}
Suppose by contradiction that this fails.
Then there is a sequence of $\omega$-compatible almost complex structures $I_n$ agreeing with $I$ outside of the compact subset $K$, and a sequence $u_n$ of Floer trajectories for $(H_{\lambda},I_n)$, with energy $E(u_n)\leq E_0$, with $F(x_{n,-})<F(x_{n,+})$ (here $x_{n,\pm}$ are the $1$-orbits of $H_\lambda$ to which $u_n$ exponentially converges by virtue of having finite energy).
As there are finitely many filtration values for the $1$-orbits of $H_{\lambda}$, we deduce that $F(x_{n,-})-F(x_{n,+})\leq e_0$ for some $e_0<0$.
As the initial points $x_{n,\pm}(0)$ of these orbits lie in a compact subset of $Y$, by passing to a subsequence we may assume that $x_{n,\pm}\to x_{\pm}$ converge to $1$-periodic orbits $x_{\pm}$ of $H_{\lambda}$.
By Gromov compactness, the Floer trajectories $u_n$ converge in $C^{\infty}_{\mathrm{loc}}$ to a broken Floer trajectory $u$ of $(H_{\lambda},I)$ with ends $x_{\pm}$, possibly with finitely many $I$-holomorphic spheres attached, and with $E(u)\leq E_0$. In particular, $F(x_{n,-})-F(x_{n,+})\leq e_0<0$ converges to the inequality $F(x_{-})-F(x_{+}) \leq e_0 <0$ which contradicts \cref{FiltrationByF}, $F(x_-)\geq F(x_+)$, for the unperturbed Floer solution $u$ of $(H_{\lambda},I)$.

The second claim is a consequence of the standard transversality argument [44, 16], independently of the issue of filtrations.

The final claim is proved analogously: now $u_n$ are continuation solutions for $(H_{\lambda},I_{s,n})$ with $I_{s,n}\to I$ as $n\to \infty$, but the Gromov limit $u$ is still a broken bubbled trajectory $u$ of $(H_{\lambda},I)$ with $I$-bubbles.
\end{proof}

\subsection{Energy-bracketed Floer cohomology}\label{Subsection Filtered Floer cohomology}

The energy bound $E_0$ is essential in \cref{Lemma perturb I subject to energy bound}. A priori, the size of the perturbation $I$ needed to achieve the filtration estimate may decay to zero as we let $E_0$ grow. There is the danger of arbitrarily high-energy conspiring Floer solutions which ``disappear'' in the limit, meaning: there may exist Floer solutions $u_n$ of $(H_{\lambda}, I_n)$, with $E(u_n)\to \infty$ and $F(x_{n,-})-F(x_{n,+})<0$, which fail to have a subsequence admitting a Gromov limit.
We circumvent this issue by imposing an artificial bound on the energy by the following algebraic trick. The construction we carry out is essentially that of ``filtered Floer cohomology'' introduced by Ono\footnote{Ono's paper uses capped $1$-periodic orbits, and works with the Floer action functional on a cover of the free loopspace. We could have done the same (for a non-contractible $1$-orbit $x: S^1 \to Y$, the analogue of a capping is then a choice of homotopy between $x$ and a chosen representative loop in the free homotopy class $[x]\in [S^1, Y]$). We opted for bypassing these matters since we are using a simpler less refined version of the Novikov ring compared to [33].} in \cite[Sec.3]{Ono}.

Initially, let us ignore the almost complex structure.
To simplify the discussion, we assume that we use the Novikov field $\k$ from \cref{EqnSec2NovikovField} (e.g.\,when $c_1(Y)=0$).
Define a modified Novikov ring
\k_{E_0} = \k_{\geq 0}/T^{E_0}\k_{\geq 0}
where $\k_{\geq 0} = \{ \sum n_j T^{a_j}: a_j\geq 0\in\R, a_j\to \infty,n_j\in\mathbb{B}  \}$. The Floer complex
is the $\k_{E_0}$-module generated by certain $1$-orbits of $H_{\lambda}$ (if one uses a cascade model, the auxiliary Morse functions chosen on the {\MB } manifolds of $1$-orbits select a subset of $1$-orbits, see \cref{AppendixCascades}). The Floer differential acts as a matrix with $\k_{\geq 0}$ entries, since the Novikov weights $T^{E(u)}$ in the counts involve nonnegative energies $E(u)\geq 0$. The matrix is therefore well-defined
also over $\k_{E_0}$ (and still squares to zero), which is tantamount\footnote{Indeed even in the proof that the differential squares to zero we ignore higher energy solutions, because the proof involves looking at broken solutions arising as Gromov limits when compatifying  $1$-dimensional moduli spaces of Floer solutions of a specified energy value, and the (nonnegative) energy of pieces of the broken solution adds up to that value.}
to ignoring Floer trajectories of energy greater than $E_0$. The natural quotient map $\k_{E_0}\leftarrow \k_{E_1}$ for $E_0\leq E_1$ induces a natural chain map
\begin{equation}\label{Eqn increase energy bracket}
C_{E_0}^*\leftarrow C_{E_1}^*.
\end{equation}
The inverse limit $C^*:=\varprojlim C^*_{E}$ over these maps as $E\to \infty$ can be identified with $CF^*(H_{\lambda},\k_{\geq 0})$.
Observe that $\k$ is the localisation of $\k_{\geq 0}$ at $T$, and the localisation $C^*\otimes_{\k_{\geq 0}} \k$ of $C^*$ at $T$ can be identified with $CF^*(H_{\lambda},\k)$. The exact functor of localisation induces an isomorphism on cohomology,
HF^*(H_{\lambda},\k) \cong H^*(C^*,\k_{\geq 0})\otimes_{\k_{\geq 0}} \k.
We need to refine the above argument, by taking into account the almost complex structure.
For now, we assume that a choice has been made of some admissible Hamiltonians $H_{\lambda}$ labelled by a sequence of the generic slopes $\lambda=\lambda_i>0$ which diverge to infinity (later we will compare two such choices). We assume that $H_{\lambda},\phi$ are constructed as in \cref{Cor Subcomplex Trick}, so that for $\lambda_i< \lambda_{i+1}$ the Hamiltonians $H_{\lambda_i},H_{\lambda_{i+1}}$ agree up to the linearity interval $L_i$.

In particular, this means that an admissible monotone homotopy $H_s$ from $H_{\lambda_{i+1}}$ to $H_{\lambda_i}$ exists (see \cref{Definition monotone hpy}), satisfying the more general condition in \cref{Lemma Floer continuation maps are ok}.

Let $\mathcal{I}_{\lambda}$ denote the space (with the $C^{\infty}$-topology) of $\omega$-compatible almost complex structures on $Y$ equal to $I$ outside of a chosen compact subset $K_{\lambda}\subset Y$ containing all $1$-periodic orbits of $H_{\lambda}$ in its interior, and equal to $I$ over all\footnote{It suffices that this holds on specified subintervals of the $L_i$, and that in arguments where we need $L_i$ to be large (in monotonicity lemma arguments on $B$) we also ensure that such subintervals of $L_i$ are large. We recall that to ensure transversality for Floer solutions, the genericity of $I'$ just needs to be ensured in some region that the solution crosses through. For example, one might wish to perturb $I$ near the boundary of the linearity interval, so that transversality is ensured for any Floer solution that tries to enter the linearity region.} of the linearity intervals $L_i$.

Let $\mathcal{I}_{E,\lambda}\subset \mathcal{I}_{\lambda}$ denote a path-connected neighbourhood of $I$ consisting of $I'$ for which all Floer solutions of $(H_{\lambda},I')$ of energy at most $E$ satisfy the filtration inequality, and all Floer continuation solutions of $(H_{\lambda},I_s)$ for any homotopy $I_s\subset \mathcal{I}_{E,\lambda}$ of energy at most $E$ satisfy the filtration inequality (that such a neighbourhood exists is implied by \cref{Lemma perturb I subject to energy bound}). When we refer to ``generic'' $I'$ or $I_s$ we are referring to the Baire second category subspaces consisting respectively of those $I'$ for which transversality holds for Floer trajectories of $(H_{\lambda},I')$ and those $I_s$ for which transversality holds for continuation maps of $(H_{\lambda},I_s)$ (here $I_s$ need not be generic for each $s$, but we assume that $I_s=I_-$ for $s\ll 0$ and $I_s=I_+$ for $s\gg 0$ for generic $I_\pm$). Denote by $\mathcal{I}_{\epsilon}\subset \mathcal{I}_{\lambda}$ all (or any collection of) neighbourhoods of $I$ labelled by partially ordered parameters $\epsilon$ (e.g.\,$\epsilon$ can be taken to equal the neighbourhood itself, partially ordered by inclusion). We abusively write ``$\epsilon\to 0$'' to mean that we are taking the directed limit over inclusions of the $\mathcal{I}_{\epsilon}$ as the $\mathcal{I}_{\epsilon}$ converge to $I$ in the $C^{\infty}$-topology.

\begin{cor}\label{Corollary defining energy filtered Floer cx}
For generic $I'\in \mathcal{I}_{E,\lambda}$, the Floer complex
is well-defined and the differential respects the filtration. The quotient $\k_{E} \leftarrow \k_{E'}$ for $E\leq E'$ induces an ``energy-restriction map'' for any $I'\in  \mathcal{I}_{E',\lambda}$,
\psi_{E,E'}:CF^*_{E,\lambda}(I') \leftarrow CF^*_{E',\lambda}(I').
Any generic homotopy $I_s\subset \mathcal{I}_{E,\lambda}$ from $I_-$ to $I_+$ induces a filtration preserving continuation map
\varphi_{I_s}: CF^*_{E,\lambda}(I_+) \to CF^*_{E,\lambda}(I_-),
which is a quasi-isomorphism.

For any generic $I_s\subset \mathcal{I}_{E',\lambda}$ and $E\leq E'$ these maps fit into a commutative diagram,
\xymatrix{
\ar@{<-}_-{\psi_{E,E'}}[d]
\ar@{->}^-{\varphi_{I_s}}[r] &
\ar@{<-}^-{\psi_{E,E'}}[d]
\\
\ar@{->}^-{\varphi_{I_s}}[r] &
Given an admissible monotone homotopy $H_s$ from $H_{\lambda_-}$ to $H_{\lambda_+}$, there is a neighbourhood $\mathcal{I}_{E,\lambda,\lambda'}\subset \mathcal{I}_{E,\lambda'}$ of $I$ for which the continuation map
\varphi_{H_s}: CF^*_{E,\lambda_+}(I') \to CF^*_{E,\lambda_-}(I').
is filtration preserving. By making the linearity intervals sufficiently large in the construction of the $H_{\lambda}$, the $\varphi_{H_s}$ will be inclusions of subcomplexes. The $\varphi_{H_s}$ maps commute with $\psi_{E,E'}$, and up to quasi-isomorphism they commute with $\varphi_{I_s}$.

In particular, for any given slopes and energies, any sufficiently small $\epsilon$ will ensure that the above statements hold for all $I'$, respectively $I_s$, inside $\mathcal{I}_{\epsilon}$.
\end{cor}
\begin{proof}
These claims follow essentially immediately from \cref{Lemma perturb I subject to energy bound}, so we just make some comments.

Regarding the energies, once we impose an energy bound $E_0$, then the proof that the Floer continuation map is a chain map over $\k_{E_0}$
only requires looking at $1$-dimensional moduli spaces of Floer continuation solutions of total energy up to $E_0$, and when such solutions break they can only give rise to a Floer trajectory of energy at most $E_0$ since energy is additive.

The commutativity of the diagram follows from the fact that both horizontal maps are induced by the natural quotient map $\k_{\geq 0} \to \k_E=\k_{\geq 0}/T^E\k_{\geq 0}$ applied to the continuation map $\varphi_{I_s}: CF^*(H_{\lambda},I_+,\k) \to CF^*(H_{\lambda},I_-,\k)$ (this map is well-defined but may not preserve the filtration). The same reasoning implies that $\varphi_{H_s}$ commutes with $\psi_{E,E'}$. That $\varphi_{H_s}$ and $\varphi_{I_s}$ commute up to quasi-isomorphism is a standard fact about Floer continuation maps.

Regarding $\varphi_{H_s}$: by \cref{Lemma perturb I subject to energy bound}, for $I'$ close to $I$, all continuation solutions respect the filtration. We then run the same proof as in \cref{Cor Subcomplex Trick} (in particular see \cref{Remark montone hpy is ok}), using that $I_s=I$ on the linearity intervals, so the monotonicity lemma for $B$ holds there.

The final statement is the observation that given $E,\lambda$ we have $I\in \mathcal{I}_{\epsilon}\subset \mathcal{I}_{E,\lambda}$ for all ``small $\epsilon$''.
\end{proof}

\subsection{Period-bracketed Floer cohomology}\label{Subsection Period-filtered Floer cohomology}

\begin{de}
Let $CF^*_{E,\lambda,[0,b]}(I')\subset CF^*_{E,\lambda}(I')$ be the subcomplex generated by all $1$-orbits whose associated $S^1$-periods lie in $[0,b]$. By \cref{Corollary defining energy filtered Floer cx} this is well-defined (see also \cref{Subsection Dependence of the filtration on phi}, and notice here we filter by period, rather than by $F$, so imposing ``period$\,\leq b$'' yields a subcomplex).
Its cohomology
$HF^*_{E,\lambda,[0,b]}(I')$ does not depend up to isomorphism on the choice of $I'\in \mathcal{I}_{E,\lambda}$. Similarly, one can define the quotient complex $$CF^*_{E,\lambda,(a,b]}(I'):=CF^*_{E,\lambda,[0,b]}(I')/CF^*_{E,\lambda,[0,a]}(I'),$$ and $HF^*_{E,\lambda,(a,b]}(I')$ will be independent of such $I'$.
We call $(a,b]$ the {\bf period filtration bracket}. If in the quotient we used $[0,a)$ instead of $[0,a]$, we would get period bracket $[a,b]$, etc.

Any filtration preserving map (see \cref{Definition filtration meaning})
induces a map with period brackets imposed, e.g.\,the commutative diagram in \cref{Corollary defining energy filtered Floer cx} holds with any period bracket.
\end{de}

\begin{rmk}
It is more intrinsic that the labels $[a,b]$ refer to $S^1$-period values $T_{-i}$ (and period zero for the constant orbits), rather than using $F$-filtration value brackets. This is because the $F$-values change if we choose a different construction of the $H_{\lambda}$ (see Section \ref{Subsection Dependence of the filtration on phi} and \cref{FiltrValue}).
\end{rmk}

\begin{cor}\label{Cor HF and LES of filtered HF}
Let $\mathcal{P}$ be any period bracket, e.g.\,$[a,b]$ or $(a,b]$ for $a,b\in [0,\infty]$.
Up to filtration preserving Floer continuation isomorphisms, the $\k_E$-modules
HF^*_{E,\lambda,\mathcal{P}}:= \varinjlim_{I'\to I} HF^*_{E,\lambda,\mathcal{P}}(I'):= \varinjlim_{\epsilon\to 0} \{ HF^*_{E,\lambda,\mathcal{P}}(I'): I'\in \mathcal{I}_{\epsilon}\}
are independent of the choices of $\mathcal{I}_{E,\lambda}$ or the homotopies $I_s$. These fit into a commutative diagram,
\xymatrix{
\ar@{<-}_-{\psi_{E,E'}}[d]
\ar@{->}^-{\varphi_{H_s}}[r] &
\ar@{<-}^-{\psi_{E,E'}}[d]
\\
\ar@{->}^-{\varphi_{H_s}}[r] &
thus the following $\k$-module is well-defined
HF^*_{\lambda,\mathcal{P}}:=\k\otimes_{\k\geq 0}\varprojlim_{E\to \infty} HF^*_{E,\lambda,\mathcal{P}}.
Up to a filtration-preserving isomorphism induced by Floer continuation maps, this $\k$-module does not depend on the choices of $H_{\lambda}$.
\end{cor}
\begin{proof}
This mostly follows by \cref{Corollary defining energy filtered Floer cx}, up to some additional comments.

Continuation maps only depend on the choice of homotopies up to chain homotopy, so the maps $\varphi_{I_s}$ and $\varphi_{H_s}$ on cohomology are independent of the choice of generic homotopies $I_s$ and $H_s$. By virtue of the colimit definition, $HF^*_{E,\lambda,\mathcal{P}}$ is  independent of the choices of $\mathcal{I}_{E,\lambda}$.

Recall that continuation maps compose correctly at the cohomology level, indeed one can work with homotopies of the pairs $(H_s,I_s)$; and composites of such maps up to chain homotopy agree with the continuation map induced by concatenating the homotopies of those pairs. Thus the maps $\varphi_{H_s}$ and $\varphi_{I_s}$ (and, as discussed above, also $\psi_{E,E'}$) all fit into filtration-preserving commutative diagrams on cohomology whenever they are defined.

Consider now two different choices of constructions of admissible Hamiltonians $H_{\lambda_1}^{(1)}$, $H_{\lambda_2}^{(2)}$ (where $\lambda_1$ ranges over some cofinal collection of values, and so does $\lambda_2$). To build an isomorphism between the two colimits that these two choices induce, it suffices to construct filtration-preserving continuation maps%
\footnote{The inverse map on direct limits will be constructed similarly via maps $HF^*(H_{\lambda_1}^{(1)},I',\k_E) \to HF^*(H_{\lambda_2'}^{(2)},I',\k_E)$ when $\lambda_1\leq \lambda_2'$. The fact that the composites of these two maps is the identity map follows because the composites $HF^*(H_{\lambda_2}^{(2)},I',\k_E) \to HF^*(H_{\lambda_1}^{(1)},I',\k_E) \to  HF^*(H_{\lambda_2'}^{(2)},I',\k_E)$ are continuation maps $HF^*(H_{\lambda_2}^{(2)},I',\k_E) \to  HF^*(H_{\lambda_2'}^{(1)},I',\k_E)$ which are part of the colimit definition, so they actually define the identity map on the colimit.}
$HF^*(H_{\lambda_2}^{(2)},I',\k_E) \to HF^*(H_{\lambda_1}^{(1)},I',\k_E)$ when $\lambda_2\leq \lambda_1$ and $I'$ is sufficiently close to $I$. As $\lambda_2\leq \lambda_1$, we can always build such a filtration-preserving admissible monotone continuation map by \cref{Cor continuation is always possible}, and being continuation maps they will automatically be compatible with the $\varphi_{I_s}$-continuation maps on cohomology, and admissibility ensures they count Floer continuation solutions with non-negative $T$ powers thus they are also compatible with the $\psi_{E,E'}$-maps.
\end{proof}

\subsection{Period-bracketed symplectic cohomology}\label{Subsection Period-filtered symplectic cohomology}
Morally, we have filtered the Floer cohomology of $(Y,\varphi)$ by ``action brackets'' induced by the action functional on the projected space $B$ via $\Psi$. This is not entirely accurate, as we use a cut-off function $\phi$ to ignore the contributions of the action functional in certain regions. The natural inclusion/truncation of action brackets induces a long exact sequence:

\begin{cor}\label{Cor long exact sequence period brack}
For $a\leq b\leq c$ in $[0,\infty]$, there is a long exact sequence of $\k$-modules
\cdots
\to
\to
\to
\to
\to
\cdots
and its $\lambda$-colimit induces a long exact sequence on period-bracketed symplectic cohomology groups,
\cdots
\to
\to
\to
\to
\to
\cdots
In the case $a=0$, $b=0$, $c=+\infty$, we get the usual long exact sequence relating the $\k$-modules $QH^*(Y)$, $SH^*(Y,\Fi)$, and $SH_+^*(Y,\Fi):=SH^*_{(0,\infty]}(Y,\Fi).$ \qed
\end{cor}

\begin{de} The Corollary ensures that the following $\k$-module is well-defined,
SH^*_{[0,b]}(Y,\Fi):=\varinjlim_{\lambda\to \infty} HF^*_{\lambda,[0,b]}=
\varinjlim_{\lambda\to \infty} \left(\k\otimes_{\k\geq 0}
\varprojlim_{E\to \infty}
\varinjlim_{I'\to I} HF^*_{E,\lambda,[0,b]}(I')\right).
\end{de}

\begin{cor}
$SH^*_{[0,b]}(Y,\Fi)$ is a graded-commutative unital $\k$-algebra admitting a natural unital $\k$-algebra homomorphism
$QH^*(Y)\to SH^*_{[0,b]}(Y,\Fi)$.
\end{cor}
\begin{proof}
That the filtration is compatible with (a careful construction of) the pair-of-pants product was shown in [41].
For $\lambda''\geq 2 \,\mathrm{max}(\lambda,\lambda')$, the pair-of-pants product partly respects the period filtration: for $a,b,a',b'\in \R\cup \{\pm \infty\}$,
$$HF^*_{\lambda,(a,b]}\otimes HF^*_{\lambda',(a',b']} \to HF^*_{\lambda'',(\mathrm{max}(a+b',a'+b),\;b+b']}$$
(by convention $a+b'=-\infty$ if $a=-\infty$, and similarly for $a'+b$). For $a=a'=-\infty$, we obtain the claimed product structure on $SH^*_{[0,b]}(Y,\Fi)$.
The natural map from $QH^*(Y)$ arises from the PSS-isomorphism $QH^*(Y)\cong HF^*_{\lambda,[0,0]}\cong HF^*_{\textrm{small slope}}$ (the latter's chain level generators are the $1$-periodic orbits of $H_{\lambda}$ with filtration value $F=0$), followed by the natural map into the direct limit.
\end{proof}

Letting $b=\lambda$ be an admissible slope for which we chose $H_{\lambda}$ in the construction, we deduce that
SH^*_{[0,\lambda]}(Y,\Fi)\cong HF^*_{\lambda}.
More generally, if $b=\mu$ equals an $S^1$-period value $T_{-i}=c'(H)=\mu\leq \lambda$, then
HF^*_{\lambda,[0,b]} \cong HF^*_{\mu^+} \cong SH^*_{[0,\mu^+]}(Y,\Fi)
where $\mu^+$ is any admissible slope\footnote{or explicitly use $H_{\mu^+}$ constructed from $H_{\lambda}$ by extending with slope $\mu^+$ in the region at infinity where $H_{\lambda}$ has slope$\,\geq\mu^+$. The filtration bracket $[0,b]$ means we ignore the new $1$-orbits that $H_{\lambda}$ has which $H_{\mu^+}$ does not, and that $CF^*_{\mu^+}(I')\subset CF^*_{\lambda}(I')$ is a subcomplex for $I'$ sufficiently close to $I$.} in the interval $(T_{-i},T_{-(i+1)},)$.

Under these identifications, a continuation map $HF^*_{\lambda}\to HF^*_{\lambda'}$ for $\lambda\leq \lambda'$ corresponds to the natural map that enlarges the right-bracket,
SH^*_{[0,\lambda]}(Y,\Fi) \to SH^*_{[0,\lambda']}(Y,\Fi).
\subsection{The $\Fi$-filtration revisited}\label{Subsection The fi filtration revisited}
We can now rewrite \cref{Definition fi-filtration on QH} as follows.

\begin{cor}
The $\Fi$-filtration ideals of $QH^*(Y)$ are
F^{\varphi}_\lambda \;\;:=\bigcap_{\mathrm{generic}\,\mu>\lambda} \ker (QH^*(Y) \to SH^*_{[0,\mu]}(Y,\Fi)).
\qquad \qed
\end{cor}

\begin{rmk}\label{Remark limits and colimits}
If one wished to carry out a more refined construction at the chain level, one could try to apply the telescope model methods by Varolgunes [45].

We comment on the order of the limits/colimits.
The colimit over $I'$ must be done first, otherwise the filtration estimates may fail. In general, the order of the $\lambda$-colimit and the $E$-limit in the definition of $SH^*(Y,\Fi)$ is important.
Having the $\lambda$-colimit on the outside means that any given class can be realised for some large enough slope $\lambda$, and a representative is described by following a ``zig-zag'' diagram
\xymatrix{
\ar@{->}^-{\varphi_{I_s}}[r]
\ar@{->}^-{\varphi_{I_s}}[r]
\cdots
\ar@{->}^-{\varphi_{I_s}}[r]
\ar@{<-}^-{\psi_{E,E'}}[d]
\\
\ar@{->}^-{\varphi_{I_s}}[r]
\cdots
where one can keep lifting the cycle to higher-energy brackets, such as $E'\geq E$ above, at the cost of applying enough quasi-isomorphisms to the cycle to reach an almost complex structure close enough to $I$, such as $I^{(N)}$ above. If however the $E$-limit were on the outside, then it may happen that a ``zig-zag'' representative requires repeated increases of $\lambda$, as it may otherwise not be possible to find a lifted \emph{cycle}. Indeed, a cycle $c$ for $(H_{\lambda},\k_E)$ will lift to a cycle $c+T^E c'$ for $(H_{\lambda},\k_{E'})$ precisely if the new boundary terms of order $T^e$, with $e\in [E,E']$, appearing in $\partial c$ can be cancelled by $T^E \partial c'$ for some $\k_{\geq 0}$-combination of $1$-orbits of $H_{\lambda}$. Increasing $\lambda$ to $\lambda'$ means that more $1$-orbits are at our disposal to build $c'$.
A simple situation, where lifting cycles is not a problem, occurs when no new $1$-orbits appear in degree $\deg (c)$ for large enough $\lambda$.
\end{rmk}

\begin{cor}
For all symplectic $\C^*$-manifolds with $c_1(Y)=0$ (e.g.\,all CSRs), for any period bracket $\mathcal{P}$ we may interchange the following (co)limits,
\k\otimes_{\k\geq 0}
\varinjlim_{\lambda\to \infty}
\varprojlim_{E\to \infty}
\varinjlim_{I'\to I} HF^*_{E,\lambda,\mathcal{P}}(I')
\cong
\k\otimes_{\k\geq 0}
\varprojlim_{E\to \infty}
\varinjlim_{\lambda\to \infty}
\varinjlim_{I'\to I} HF^*_{E,\lambda,\mathcal{P}}(I').
Given a degree $k$, for $\lambda$ sufficiently large, and any $a,b\in \R\cup \{\pm \infty\}$, we can identify
\cong
HF^k_{\lambda,\mathcal{P}} := \k\otimes_{\k\geq 0}
\varprojlim_{E\to \infty}
\varinjlim_{I'\to I} HF^*_{E,\lambda,\mathcal{P}}(I').
In particular, picking $\lambda$ large enough so that the above holds in all degrees $k\in [-1,2\dim_{\C} Y)$, then for $I'$ sufficiently close to $I$ we have a $\k$-isomorphism
\partial: HF^{k-1}_{\lambda,(0,\infty]}(I') \stackrel{\cong}{\longrightarrow} QH^k(Y;\k)
induced by the Floer differential,\footnote{The image of $\partial(c)$ on a class $[c]\in HF^{k-1}_{\lambda,(0,\infty]}(I')$ lies in filtration zero, since $[\partial c]=0\in HF^{k-1}_{\lambda,(0,\infty]}(I')$.} for $k\in [0,2\dim_{\C}Y)$ (i.e.\,where $QH^k(Y;\k)$ may be supported).

The $\Fi$-filtration ideals defined by \cref{Definition fi-filtration on QH} on $QH^*(Y;\k)$ are then precisely the images
\partial: HF^{*-1}_{\lambda,(0,b)}(I') \longrightarrow QH^*(Y;\k)
as $b>0$ varies (this image can change only for outer $S^1$-period values $b$, which are non-admissible slopes $\lambda$, where the $\Fi$-filtration of \cref{Definition fi-filtration on QH} can jump).
\end{cor}
\begin{proof}
The fact that the localisation $\k \otimes_{\k_{\geq 0}}$ commutes with the $\lambda$-colimit
just uses the categorical fact that two colimits always commute (localisation at $T$ of a $\k_{\geq 0}$-module $M$ can be viewed as a colimit over inclusions of the modules $\k_{\geq  a}\otimes_{\k_{\geq 0}} M$ as $a \to -\infty$).
By \cref{PropSec2RSIndices},
in any given degree, there are only finitely many generators at the Floer chain level in that degree ($1$-orbits of $H_{\lambda}$ corresponding to the critical points of the auxiliary Morse functions on {\MB } manifolds of $1-$orbits of $H_{\lambda}$, for $\lambda$ sufficiently large, see \cref{AppendixCascades} -- see also \cref{Rmk perturbns of Ham issue}).
So the condition mentioned at the end of \cref{Remark limits and colimits} ensures that the $\lambda$-colimit and $E$-limit commute.

As the chain complex in a given degree $k$ is finitely generated, the $\lambda$-colimit is taken over linear maps $\varphi_j$ of finite dimensional $\k$-vector spaces $V_j$ of uniformly bounded dimension. In such a general situation, $V_1 \stackrel{\varphi_1}{\to} V_2 \stackrel{\varphi_2}{\to} V_3 \to \cdots$, by considering the dimension of the direct limit, there must be a finite $n$ such that all classes are represented in $V_n$, and the direct limit is naturally identifiable with $V_n/\ker \varphi_n^m$ where we pick any large enough $m$ since the kernel of $\varphi_n^m:=\varphi_{n+m}\circ \varphi_{n+m-1}\circ \cdots \circ \varphi_{n}$ stabilizes for dimension reasons.
In our case, the $\varphi_i$ are the cohomology maps of inclusions of subcomplexes, and $V_n$ is the Floer cohomology for $H_{\lambda}$ for large enough $\lambda$. We may assume that increasing $\lambda$ further does not produce new $1$-orbits in degrees $k$ and $k-1$, thus $\varphi_n^{m}$ is the identity map on cohomology.

The final claim follows from the long exact sequence in \cref{Cor long exact sequence period brack}, since a class in $QH^*(Y)$ vanishes via the natural map $\widetilde{c}_b:QH^*(Y)\cong HF^*_{\lambda,[0,0]} \to HF^*_{\lambda,[0,b]}$ if and only if it lies in the image of the connecting map $HF^{*-1}_{\lambda,(0,b]}\to HF^*_{\lambda,[0,0]}$ of the long exact sequence (and by chasing the definition of the connecting map of a cohomology long exact sequence induced by a short exact sequence one sees that the connecting map is the Floer differential before quotienting out any high-filtration terms). By the final claim in \cref{Cor HF and LES of filtered HF}, if $b=T_{-i}=c'(H)=\mu$, then $\widetilde{c}_b$ can be identified with the map $c_{\mu^+}$ used in \cref{Definition fi-filtration on QH}.
\end{proof}

\begin{rmk}[Perturbations of the Hamiltonian]
\label{Rmk perturbns of Ham issue}
In the above discussion we used the {\MB } (i.e.\,cascade) model for Floer cohomology (\cref{AppendixCascades}), as we did not perturb $H_{\lambda}$. One could carry out a perturbation $H_{\lambda}'$ of $H_{\lambda}$ near the {\MB } manifolds to obtain non-degenerate orbits, keeping in mind that this ruins the filtration%
\footnote{\eqref{EqnPhiProjectionNice} fails if we perturb $H_{\lambda},$ one gets a non-vanishing term $d\a(\xi,W)$ in equation \eqref{omegaV}, where $W:=\Psi_*(X_{\widetilde{H}_\lambda}-X_{H_\lambda})$ is the projection of the difference in Hamiltonian vector fields in $Y$ that occurred by perturbation. The change in the difference of the filtration values \eqref{F_DifferenceInTermsOfOmega} is caused by integration $\int_{-\infty}^{+\infty} d\a(\partial_s v,W)ds$ of this term, which we cannot estimate.}
analogously to what happened for perturbations of $I$. By using the perturbation procedure described in \cref{Appendix {\MBF } theory: perturbations}, one can ensure that the non-constant $1$-orbits are a subset of the original $1$-orbits and therefore project via $\Psi$ to $S^1$-orbits in $B$, so their filtration values are unchanged. The same Gromov compactness arguments used for perturbations $I'$ of $I$ apply when we use perturbations $H_{\lambda}'$ of $H_{\lambda}$, and similarly for continuation maps we use perturbations $I_s',H_s'$ of $I,H_s$. Thus the same arguments as above hold after replacing the innermost $\varinjlim$ over $I'\to I$ by a $\varinjlim$ over pairs $(I',H') \to (I,H)$.
\end{rmk}

\subsection{Adapting energy-bracketing to the $S^1$-equivariant setup}\label{Subsection Filtered Floer cohomology equiv setup}

In \cref{Subsection intro S1 spectral seq}
we work with $S^1$-equivariant Floer cohomology in the sense of [28], where a $1$-orbit contributes a copy of the $\k[u]$-module $\mathbb{F}$, rather than $\k$, where
$\mathbb{F}=\k(\!(u)\!)/u\k [\![u]\!]\cong H_{-*}(\C\P^{\infty})$.
In \cref{Subsection Filtered Floer cohomology} we explained energy-bracketing using $
\k_{E_0} = \k_{\geq 0}/T^{E_0}\k_{\geq 0}
$, so that each $1$-orbit contributed to $C^*_{E_0}:=CF^*(H_{\lambda},\k_{E_0})$ a copy of  $\k_{E_0}$. In the $S^1$-equivariant setup, we instead get a contribution of $\mathbb{F}_{E_0}$ for each $1$-orbit, where
\mathbb{F}_{E_0}:=
\k_{E_0}(\!(u)\!)/u\k_{E_0} [\![u]\!].
The non-equivariant methods then adapt very easily to the $S^1$-equivariant case, and will be omitted.
\section{Morse--Bott--Floer spectral sequences}\label{SpecSeqMBF}
\newcommand{\RZ}[1]{Y_{#1}}

\subsection{Morse--Bott manifolds of Hamiltonian 1-orbits of $H_\lambda$}\label{SubsectionHam1-orbitsOfHlambda}
We continue with the initial assumption from \cref{filtrationFloer}. %
Call {\bf 1-orbits} the 1-periodic orbits of the Hamiltonian $H_\lambda$ from \cref{filtrationFloer}. As $X_{H_\lambda}=c'(H)X_H,$ the 1-orbits %
are either constant $1$-orbits at points in $\F=Crit(H_\lambda)=Crit(H),$ or, for some $p=-1,\dots,-r,$ are non-constant $1$-orbits in a {\bf slice}
\begin{equation}\label{DefinionSlice}
\Slicep:=\{y \in Y \mid H(x)=H_p\}\subset  Y,
\end{equation}
where the values $H_p$ are defined by $c'(H_p)=T_p.$
Denote by $\mathcal{O}_p:=\mathcal{O}_{p,H_{\lambda}}$ the moduli space of parametrised $1$-orbits of $H_\lambda$ in $\Slicep,$ and by $B_p$ the set of their initial points, $$B_p:=\{x(0)\mid x\in \mathcal{O}_p \}\subset \Slicep.$$
It will be convenient to allow the notation in the case $p=0$, for the constant $1$-orbits $B_0=\mathcal{O}_0=\F$, so $T_0=0$ (also $T_p=c'(T_p)$ will be abuse of notation when $p=0$, as $H_0$ is not well-defined since $H$ may take different values on the connected components $\F_\a$ of $\F$, and we do not require $c'=0$ there).
By \cref{FiltrValue}, the filtration value $F$ on $1$-orbits only depends on $c'(H_p)=T_p$, so let
F_p:=F(\mathcal{O}_p) \textrm{ for } p\leq 0, \textrm{ and }F_p=p \textrm{ for } p \geq 0.
Recall $X_{S^1}=X_H$ is the vector field of the $1$-periodic $S^1$-flow $\varphi_{e^{2\pi it}}$, so $t\in \R/\Z \cong S^1$.

Recall $Y_m$ denotes the set of $\Z/m$-torsion points in $Y$ (\cref{TorsionPtsTorsionSubmflds}), so these points are fixed points of the $S^1$-flow for time $1/m$.

\begin{lm}\label{BpcAreTorsionIntersectedBySlice}
$\displaystyle B_p= \Sp \cap \RZ{m}$ %
where $T_p=k/m$ for coprime $k,m\in\N.$
\end{lm}
\begin{proof}
$B_p$ is the union of the  period $T_p$ orbits of the $S^1$-flow on $\Sp$, as
$X_{H_\lambda}=c'(H_p)X_H = T_p X_{S^1}$ on $\Sp.$
As all proper closed subgroups of $S^1$ are discrete,
$T_p$ has to be a rational number, $T_p=k/m$ for some coprime $k,m\in\N.$
The latter condition, $(k,m)=1,$ implies $\a k + \b m=1$ for some $\a,\b\in\Z$.  Thus, points in $B_p$ are also fixed by the  $S^1$-flow for time
$\a k/m = (1/m)- \b$, and thus also for time $1/m$, so they lie in $\RZ{m}.$ Thus, $B_p \subset \Sp \cap \RZ{m}$.
Conversely, a point in $\Sp \cap \RZ{m}$ is fixed by the  $S^1$-flow for time $1/m$, and thus also for time $T_p$, so it lies in $B_p.$
\end{proof}

\begin{cor}\label{Cor dim of Bpbeta}
The connected components of $B_p=\Sp \cap Y_m$ are precisely the smooth submanifolds
\begin{equation}\label{EqConnCompOfBp}
B_{p,\c}=\Sp \cap Y_{m,\c},
\end{equation}
where the $\beta$-labelling runs over the non-compact connected components $Y_{m,\b}$ of $Y_m$.

\end{cor}
\begin{proof}
By \cref{BpcAreTorsionIntersectedBySlice}, $B_p$ is the fixed locus of the $\Fi$-action by $\Z/m\leq \C^*$ on $\Sp,$ so it is a smooth submanifold of $\Sp$.\footnote{being a fixed locus of a compact Lie group action on a smooth manifold, \cite[p.108]{DK00}.}
By \cref{TorsionMfdIsSymplectic}, $Y_{m,\c}\subset Y$ is a symplectic $\C^*$-submanifold so \cref{FibersMomentMapConnected} applied to $H|_{Y_{m,\c}}: Y_{m,\c} \to \R$ shows that
$H|_{Y_{m,\c}}^{-1}(H_p)=\Sp \cap Y_{m,\c}$ is connected.
Observe that only non-compact components $Y_{m,\c}$ intersect $\Sp$. For $\beta\neq \beta'$, the $Y_{m,\b},Y_{m,\b'}$ are disjoint, so the connected closed submanifolds $B_{p,\b},B_{p,\b'}$ are disjoint and are thus distinct connected components of $B_p$.
\end{proof}

Moroever, the above $\beta$-labelling depends on $m,$ but not on $k$:

\begin{lm}\label{Only the primitive Ones matter}
Suppose $T_{p_1}=k_1/m$ and $T_{p_2}=k_2/m$ have the same denominators, where $(k_1,m)=(k_2,m)=1$.
Then $B_{p_1}$ and $B_{p_2}$ are diffeomorphic, indeed  diffeomorphic to the $B_{p}$ with $T_{p}=1/m.$
\end{lm}
\begin{proof}
$B_{p_1,\c} = \Sigma_{p_1} \cap \Ymc$ and $B_{p_2,\c} = \Sigma_{p_2} \cap \Ymc,$ we have
$$B_{p_1,\c} \iso B_{p_2,\c}$$
via the normalised gradient flow $\nabla H/\norm{\nabla H}^2$ that flows from one to the other. This holds since $H$ is constant on $\Sp$, and $\Ymc$ is preserved by the gradient flow of $H$ (which is the action by $\R_+\subset \C^*$).%
\end{proof}

Consider\footnote{The orthogonal $\perp$ is calculated with respect to the metric $g(\cdot,\cdot)=\om(\cdot,I\cdot).$} the complex codimension-one distribution $\xi:=(\la X_{\R_+} \ra \oplus \la X_{H_\lambda} \ra)^{\perp}$ on $ Y\setminus \F.$ Thus, for any point $x\in  Y \setminus \F,$ we have orthogonal splittings of tangent spaces
\begin{equation}\label{splittingTangentSpaceContact}
T_x  Y= \la X_{\R_+} \ra \oplus \la X_{H_\lambda} \ra \oplus \xi,\ \qquad  \ T_x \Sp =\la X_{H_\lambda} \ra \oplus \xi.
\end{equation}

\begin{de}
Denote by $\phi_{\tau}^{X}$ the flow of a vector field $X$ for time $\tau\in \R.$ When $X=X_F$ is a Hamiltonian vector-field we
will abbreviate this flow by $\phi_{\tau}^{F}$.
\end{de}

\begin{lm}\label{LemmaForTheShear}
$(\phi_{\tau}^{{H_\lambda}})_* X_{\R_{+}}=X_{\R_+}+\tau c''(H)\norm{\nabla(H)}^2 X_{H}.$
\end{lm}
\begin{proof}
Let $\x\in Y.$ Abbreviate $\gamma(s)=\phi_s^{X_{\R_+}}(\x)=
e^{2\pi s} \cdot \x$, so $\gamma'(s)=X_{\R_+}(\gamma(s))=(\nabla H)(\gamma(s))$, and
$$(\phi_{\tau}^{{H_\lambda}})_*(\x) X_{\R_{+}}={\phi_{\tau}^{{H_\lambda}}}_*\left(\left.{\tfrac{d}{ds}}\right|_{s=0}\gamma(s)\right)=
\left.{\tfrac{d}{ds}}\right|_{s=0} (\phi_{\tau}^{{H_\lambda}}(\gamma(s)))=\left.{\tfrac{d}{ds}}\right|_{s=0}(\phi_{\tau c'(H(\gamma(s)))}^{{H}}(\gamma(s))),$$
using $X_{H_\lambda}=c'(H)X_H.$
Let $u(s,t):=\phi_t^{H}(\gamma(s))$ (which is $e^{2\pi (s+it)} \cdot x$).
The last derivative becomes: 	\begin{align*}
\left.{\tfrac{d}{ds}}\right|_{s=0} u(s,\tau c'(H(\gamma(s))))&= {\tfrac{\partial u}{\partial s} } +\left(\left.{\tfrac{d}{ds}}\right|_{s=0}\tau c'(H(\gamma(s)))\right) \tfrac{\partial u}{\partial t }
\\&= X_{\R_{+}} +\tau c''(H(x))\norm{\nabla H(x)}^2 X_{H}.
\qedhere
\end{align*}
\end{proof}

Now we prove the following non-degeneracy property of the submanifolds $B_{p,\c},$ which is analogous to the Morse-Bott non-degeneracy condition for critical submanifolds.\footnote{As we can think of $B_{p,\c}$ being the critical submanifolds for the action $1-$form $d\mathcal{A}_{H_\lambda}(x)(\xi)=\int \om(\dot{x}-X_{H_\lambda},\xi).$}

\begin{prop}\label{TorsionAreMorseBottSubmanifolds}
Given any point $\x\in B_{p,\c},$ the 1-eigenspace of the linearised return map of the flow for $H_\lambda$
is precisely the tangent space of $B_{p,\c},$ so
$\Ker\,(d_\x \phi_1^{{H_\lambda}}-Id)=T_\x B_{p,\c}.$
\end{prop}
\begin{proof}
As in Lemma \ref{BpcAreTorsionIntersectedBySlice}, let $T_p=k/m,$ for coprime $k,m\in\N.$
Then, on $\Sp$, $$\phi_1^{{H_\lambda}}=\phi_{T_p}^{{H}}=(\phi_{1/m}^{H})^k.$$
So, given $v\in T_\x \Sp,$ the condition $(\phi_1^{{H_\lambda}})_*(v)=v$ is equivalent to $(\phi_{1/m}^{H})_*^k(v)=v,$
and thus to $(\phi_{1/m}^{H})_*(v)=v,$ due to $(k,m)=1.$
Note $\phi_{1/m}^{H}$ generates a $\Z/m$-action on $\Sp$ by isometries (since the metric is $S^1$-invariant), and its fixed
locus is $B_p$ by Lemma \ref{BpcAreTorsionIntersectedBySlice}.
Thus, by \cite[Ch.VI, Thm.2.2]{Bre72}, there is a $\Z/m$-invariant tubular neighbourhood $\mathcal{N}B_{p,\c}\subset \Sp$ of $B_{p,\c}$
arising as the image of the $\Z/m$-equivariant exponential map on a neighbourhood $NB_{p,\c}$
of the zero section of the normal bundle of $B_{p,\c}$.
This identifies $NB_{p,\c}$ with $\mathcal{N}B_{p,\c}$, making $\Z/m$ act on $NB_{p,\c}$ by the linearisation $(\phi_{1/m}^{H})_*$.
Hence, if a vector $v\in T_x \Sp$ satisfies $(\phi_{1/m}^{H})_*(v)=v,$
then $v\in (N B_{p,\c})_x$ is fixed by the $\Z/m$-action and so is the curve $exp(vt) \subset \mathcal{N}B_{p,\c}.$ As $B_{p,\c}$ is an isolated fixed locus in $\mathcal{N}B_{p,\c}$, this curve has to lie in $B_{p,\c},$ so $v\in T_\x B_{p,\c}.$ Thus $\Ker((d_\x \phi_1^{{H_\lambda}}-Id)|_{T_\x \Sp})=T_\x B_{p,\c}.$
Note $\R X_{\R_+}$ is a complementary subspace to $T_\x \Sp\subset T_\x Y$, so the claim follows by Lemma \ref{LemmaForTheShear} using that $c''(H)>0$ for $c'(H)=T_p.$
\end{proof}

\begin{lm}\label{LinearizationIsComplexLinear}
The linearisation of the $S^1$-flow is complex linear with respect to a unitary trivialisation of $\xi$ along any 1-orbit in $B_{p,\c}.$
\end{lm}
\begin{proof}
Follows immediately as the $S^1$-flow is $I$-linear and a unitary trivialisation is complex-linear.
\end{proof}

\subsection{The {\MB } spectral sequence for $HF^*(H_\lambda$)}\label{SpectralSequenceConstruction}
We now build a spectral sequence analogously to \cite[Sec.7]{McLR18}. There, %
the space at infinity was a positive symplectisation, involving a radial coordinate $R$, an exact symplectic form, contact hypersurfaces $R=\mathrm{constant}$ and a Reeb flow (the Hamiltonian flow for $R$). In our setting, on $Y^{\mathrm{out}}$, we instead have the coordinate $H$, a typically non-exact symplectic form, level sets $H=\mathrm{constant}$ and the $S^1$-flow (the Hamiltonian flow for $H$).

As the Hamiltonian $H_\lambda$ from \cref{SectionConstructionOfHLambda} is autonomous, its 1-orbits are not isolated: they arise in {\MB } manifolds by \cref{SubsectionHam1-orbitsOfHlambda}, labelled $B_{p,\c}$ and $\F_\a$.  We will therefore use a \textbf{{\MB } Floer complex} $BCF^*(H_{\lambda})$ as described in \cref{AppendixCascades}, involving a choice of auxiliary Morse functions $f_{p,c}:B_{p,\c}\to \R$ and $f_{\a}:\F_\a\to \R$. To unify the notation, we write $f_i:M_i \to \R$ for these auxiliary Morse functions, denoting the {\MB } manifolds by $M_i$. As explained in \cref{AppendixCascades} we will blur the distinction between the initial points of the $1$-orbits $x_0\in M_i$ and their associated $1$-orbits $x(t)$, $x(0)=x_0$ (in \cref{SubsectionHam1-orbitsOfHlambda} we distinguished them via the notation $B_p$ and $\mathcal{O}_p$). We call {\bf critical $1$-orbits} the $1$-orbits associated to the critical points $x_0\in \mathrm{Crit}(f_i)\subset M_i$. These are the free generators of $BCF^*(H_{\lambda})$. By \cref{Lemma MorseBottFloer is perturbed Floer},
there is an isomorphism
$$BHF^*(H_{\lambda}) \cong HF^*(\widetilde{H}_{\lambda})$$
between the {\MB } Floer cohomology (for the unperturbed $H_{\lambda}$) and the Floer cohomology for any generic time-dependent compactly supported perturbation $\widetilde{H}_{\lambda}$ of $H_{\lambda}$; moreover these isomorphisms are compatible with continuation maps. That compatibility ensures that the isomorphisms behave well in the limit constructions used when we energy-bracket as in \cref{Subsection Filtered Floer cohomology}.
So for all intents and purposes we may simply write $HF^*(H_{\lambda})$ instead of $BHF^*(H_{\lambda})$. Also, up to continuation isomorphisms, there is no ambiguity in the meaning of $SH^*(Y,\varphi)$, and we may period-filter as in \cref{Subsection Period-filtered symplectic cohomology}.

We will be using the terminology introduced in \cref{AppendixCascades} (ledge, drop, crest, base, cascade, etc.). A {\bf simple cascade} is a cascade with no drops, so it is a Morse trajectory inside some $M_i$ for $f_i$.

\begin{de}
A {\bf vertical drop} is a drop whose crest and base lie in {\MB } submanifolds with the same period value $c'(H)$, i.e.\,the asymptotics lie in the same slice $\Sigma_p$ (see \cref{SubsectionHam1-orbitsOfHlambda}). A {\bf self-drop} is a vertical drop whose crest and base lie in the same Morse-Bott submanifold.
\end{de}

\begin{lm}\label{Lemma filtration for cascades}
\label{Lemma excluding vertical drops between certain Bott manifolds}
If there is a cascade solution joining critical $1$-orbits $x_-\in M_i$ to $x_+\in M_j$ for the Hamiltonian $H_{\lambda}$ and the unperturbed almost complex structure $I$, then the filtration satisfies
\hspace{20ex} F(x_-)\geq F(x_+) \qquad (\Leftrightarrow\,\textrm{the periods sastisfy }c'(H(x_-))\leq c'(H(x_+)),
with equality if and only if the cascade is simple or all of its drops are vertical drops.
In particular, $F$ remains constant on ledges and vertical drops, and vertical drops for $F\neq 0$ are as in \cref{Lemma Floer solutions for constant filtration value}.

The cascades for $I$ with $F(x_-)=F(x_+)=F_p$ are trapped in the neighbourhood of the slice $\Sigma_p$ where $\phi$ is constant, which only contains $1$-orbits from $\mathcal{O}_p$. For $F_p \neq 0$, the $\Psi$-image of such cascades lies in an $m$-torsion submanifold of $B$ (\cref{TorsionPtsTorsionSubmflds}) where $T_p=c'(H(x_{\pm}))=k/m\in \Q$ for $k,m$ coprime.
If a vertical drop $u$ has asymptotics $u_{\pm}$ in $B_{k/m,\beta_{\pm}}$,
then the $Y_{m,\beta_{\pm}}$ have overlapping $\Psi$-projections in $B^{\mathrm{out}}=\Sigma \times [R_0,\infty)$, as
$\mathrm{Image}\,(\Psi(u))\subset \Psi(Y_{m,\beta_-})\cap \Psi(Y_{m,\beta_+})\neq \emptyset$.

Given $E>0$, for any perturbation $I'$ sufficiently close to $I$ the cascades for $I'$ of energy at most $E$ are close to those for $I$ (in $C^0$ and in $C^{\infty}_{\mathrm{loc}}$) and any vertical drop $u$ from $u_{-}$ to $u_+$ for $F\neq 0$ satisfies
\begin{itemize}
\item $H(u_-)=H(u_+)$, and $u_{\pm}$ are both non-constant $1$-orbits;
\item $u_{\pm}$ correspond to orbits of the $S^1$-flow of equal period $T:=c'(H(u_\pm))$;
\item $v=\Psi(u)$ lies entirely in a region where $\phi$ is constant;
\item $v_{\pm}=\Psi(u_{\pm})$ correspond to Reeb orbits of period $T$ (but may not have the same $R$-value);
\item $\mathrm{Im}(v) $ is close to the $\C^*$-orbit of some point $p\in B$;
\item $\mathrm{Im}(v)$ is close to some $m$-torsion submanifold of $B$, where $T=k/m\in \Q$ for $k,m$ coprime;
\item $\mathrm{Im}(du)$ is close to $\C\cdot X_{S^1}\oplus \ker \Psi_*$.
\end{itemize}
Thus, if that vertical drop $u$ for $I',E$ goes from $B_{k/m,\beta_-}$ to $B_{k/m,\beta_+}$,
then the
torsion submanifolds $Y_{m,\beta_{\pm}}$ have overlapping $\Psi$-projections in $B^{\mathrm{out}}=\Sigma \times [R_0,\infty)$, so
$\Psi(Y_{m,\beta_-})\cap \Psi(Y_{m,\beta_+})\neq \emptyset$.
\end{lm}
\begin{proof}
For the first part we use \cref{H_lambdaIsOneDirected} and \cref{Lemma Floer solutions for constant filtration value}. Recall that a ledge lies in a {\MB } submanifold, and $F$ is constant there. When $F(x_-)=F(x_+)$ the ledges involve {\MB } manifolds with the same period $T_p=c'(H_p)$, so they lie in the same torsion submanifold of $Y$.
Cascades for $I$ with only vertical drops cannot exit the $\phi=\textrm{constant}$ neighbourhood of $\Sigma_p$, as it would enter the region where the filtration $1$-form is strictly negative on a drop, contradicting $F(x_-)=F(x_+)$.

The claim about $E_0,I'$ follows by the Gromov-compactness argument from \cref{Subsection Filtered Floer cohomology}, and suitably rephrasing \cref{Lemma Floer solutions for constant filtration value} after perturbation.
In more detail, in view of the energy-bracketing construction of \cref{Section Transversality for Floer solutions}, the only vertical drops for the perturbed $I'$ that get counted in the Floer complex must be close to (possibly broken and bubbled) vertical drops for the unperturbed $I$. Now $I$-holomorphic bubbles must lie entirely in the fibres of $\Psi$ as $B$ only admits constant $I_B$-holomorphic spheres. The filtration cannot detect those spheres. So a broken and bubbled vertical drop for $I$ will project to a broken Floer trajectory in $B$. In view of the zero filtration difference, it follows that all components of that Floer trajectory in $B$ are trapped in a $\C^*$-orbit.

The first claim about the $B_{k/m,\beta_{\pm}}\subset Y_{m,\beta_{\pm}}$, follows from the fact that $\Psi(Y_{m,\beta_{\pm}})$ are $\C^*$-invariant, and $\mathrm{Image}\,(v)$ lies in a $\C^*$-orbit. The second claim follows as a perturbed $v$ is close to such a solution.
\end{proof}

As the almost complex structure needs to be perturbed to achieve transversality for cascade solutions, it is understood that we run the energy-bracketing construction of \cref{Section Transversality for Floer solutions} to ensure that the filtration inequality persists. Abbreviate our {\bf period-filtered energy-bracketed Floer complex} by
using the notation from \cref{Subsection Filtered Floer cohomology} and \cref{Subsection Period-filtered Floer cohomology}, where $I'$ is a perturbation of $I$; the constant $E>0$ prescribes the energy-bracketing which prescribes the choice of Novikov ring $\k_{E}=\k_{\geq 0}/T^{E}\k_{\geq 0}$; and $\mathcal{P}\subset [0,\infty]$ is a period bracket forcing generators to have the chosen period values $T_p=c'(H_p)$. For example, $\mathcal{P}=(0,\infty]$ gives rise to the positive Floer complex where we quotiented out by the subcomplex of constant orbits.

Recall that $1$-orbits in $M_i$ have the same filtration value $F_p$ dependent only on the period value $T_p=c'(H_p)$ associated with $M_i$ (so $M_i\subset \mathcal{O}_p$), by \cref{SubsectionHam1-orbitsOfHlambda}.

\begin{cor}
The {\MBF } differential non-strictly increases the $F$-filtration $F_p=F(\mathcal{O}_p)$, equivalently it non-strictly decreases the period-filtration $T_p=c'(H_p)$.
\end{cor}
\begin{proof}
This follows by the Gromov-compactness argument from \cref{Subsection Filtered Floer cohomology}, and the filtration inequality from \cref{Lemma filtration for cascades}.
\end{proof}
Thus $C^*$ admits a {\bf filtration}, using the conventions of \cite[Sec.7]{McLR18}:
$$F^p(C^k):=\{x\in C^k \mid F(x) \geq F_p \},$$
where $\mathbf{k=p+q}$ denotes the {\bf total degree}.
In particular, $F^p(C^k)=0$ for $p>0$ since $F\leq 0$ on all $1$-orbits.
The filtration determines an associated spectral sequence $E_r^{pq}$ such that
\begin{de}
By \textbf{slope} of the $p$-th column of $E_r^{pq}$ we mean the number $c'(H_p)=T_p.$
By construction, $E_r^{pq}=0$ if the slope $T_p\notin \mathcal{P}$.
\end{de}

\begin{prop}\label{PropSpectralSeqForHFHlambda}
The spectral sequence converges,
E_r^{pq} \Rightarrow H^*(C) = HF^*_{E,\lambda,\mathcal{P}}(I'),
where $*$ will agree with the total degree $k=p+q$. It satisfies $E_r^{pq}=0$ for $p>0$. For $p=0$ and $r=1$ it recovers the quantum cohomology if $0\in \mathcal{P}$:
E_1^{0q}\cong QH^q(Y) \textrm{ if }0\in \mathcal{P} \qquad (E_1^{0q}=0 \textrm{ if }0\notin \mathcal{P}).
\end{prop}
\begin{proof}
By construction, the filtration is exhaustive and bounded below, thus it converges.
The claim follows immediately, except for the final two statements.
The condition $0\in \mathcal{P}$ means we do not quotient out by the generators with filtration value $F=0$.
These generators are precisely the constant $1$-orbits arising as the {\MB } submanifolds $\F_\a$.
Recall that the filtration ensures that these generate a subcomplex (see \cref{PositiveSH}). In particular, because the filtration difference at the ends of a cascade is zero, by \cref{Lemma filtration for cascades} and the construction of $H_{\lambda}$ in \cref{SectionConstructionOfHLambda}, the cascades are trapped in a compact region of $Y^{\mathrm{in}}$ where $c'$ was chosen to be small. It follows that $E_1^{0q}$ agrees with the Floer cohomology of a Hamiltonian of arbitrarily small slope (i.e.\,the case when $\lambda$ is smaller than any non-zero period), and recall %
this is isomorphic to $QH^*(Y)$ via a PSS-isomorphism.
\end{proof}

We now compute $E_1^{pq}$ for $p<0$.
We will now use the notation $B_p,\mathcal{O}_p$ from \cref{SubsectionHam1-orbitsOfHlambda}, and the construction of {\bf local {\MB } Floer cohomology} from
\cref{Subsection Local Floer cohomology and the Energy spectral sequence}.

\begin{lm}\label{Lemma small energy cascades near MB mfd}
Let $N_i$ be a neighbourhood of $M_i\subset Y$ disjoint from the other {\MB } submanifolds $M_j\neq M_i$.
Then cascades in $N_i$ only involve self-drops, and only involve $1$-orbits in $M_i$.

If $\omega$ is orbit-atoroidal on $M_i$ (\cref{Definition simple Bott mfds atoroidal Mi}), these cascades are all simple cascades, i.e.\,Morse trajectories in $M_i$. This holds if $\omega$ is exact on $M_i$, or if $\omega$ is aspherical on $\overline{\cup_{t\in (0,1]} \varphi_t(M_i)}$ (\cref{Remark orbit atoroidal examples}).

There is a constant $E_0>0$ such that every cascade for $H_{\lambda}$ of energy at most $E_0$ is a simple cascade.
\end{lm}
\begin{proof}
This follows by \cref{Theorem self-drop exclusion theorem} and \cref{Lemma small energy implies cascades are near Bott mfds}.
\end{proof}

\begin{cor}\label{Cor energy spectral seq}
For $p\leq 0$, and $T_p\in \mathcal{P}$,
E_1^{pq} =  BHF^*_{\mathrm{loc}}(\mathcal{O}_p,H_\lambda,I';\k_{E}).
Assume $c_1(Y)=0$. For $p\leq 0$, $T_p\in \mathcal{P}$, and for sufficiently small energy $E=E_0>0$,
E_1^{pq} =  BHF^*_{\mathrm{loc}}(\mathcal{O}_p,H_\lambda,I';\k_{E_0})
\cong \bigoplus_{i\textrm{ with } F(M_i)=F_p}
H^{*-\mu_{H_{\lambda}}(M_i)}(M_i)\otimes \k_{E_0}
is the ordinary cohomology of the components $M_i$ of $B_p$ computed as Morse cohomology for the auxiliary Morse functions, with a grading shift by
\begin{equation}\label{Equation Grading shift spectral seq}
\mu_{H_{\lambda}}(M_i) := \dim_\C\, Y - \tfrac{1}{2}\dim_{\R} M_i - RS(M_i, H_{\lambda})
\end{equation}
where $RS(M_i,H_{\lambda})$ is the Robbin-Salamon index of one (hence any) $1$-orbit in $M_i$ for $H_{\lambda}$.

If $T_p\notin \mathcal{P}$, then $E_1^{pq}=0$ by definition.
\end{cor}
\begin{proof}
The induced differential on $E_0^{pq}$ (whose cohomology is $E_1^{pq}$) only depends on cascades whose ends $x_{\pm}$ are $1$-orbits with equal filtration value $F=F_p$. By \cref{Lemma filtration for cascades}, their drops must all be vertical drops, and the cascades are trapped in a neighbourhood of $B_p$. So the local Morse-Bott Floer cohomology of $B_p$ is well-defined by \cref{Subsection Local Floer cohomology and the Energy spectral sequence}, yielding the first claim. The second claim follows by the last part of \cref{Lemma small energy cascades near MB mfd}, Theorem \ref{Prop appendix spectral sequence and E1 page} and \cref{Theorem Construction of the local system}.
\end{proof}

Consider now how the above $E_1$-pages $E_1^{pq}:=E_{1}(H_{\lambda})$ depend on the slope $\lambda$. By \cref{Cor Subcomplex Trick}, as we increase the slope $\mu$, the $E_1$-columns associated with slope values up to $\lambda$, and their edge-differentials in the spectral sequence, will not vary for slopes $\mu \geq \lambda$. Also, there are continuation maps $E_{1}(H_{\lambda})\to E_{1}(H_{\mu})$ which induce the identity map on the columns with slopes $\leq \lambda$. Thus, one can take the various (co)limits explained in \cref{Section Transversality for Floer solutions}, so that $I'\to I$, $E\to \infty$ and $\lambda \to \infty$,
implying the following corollary. Denote $H_{p,\c}^*$ the $E_{\infty}$-page of the energy-spectral sequence, whose $E_1$-page is the local Floer cohomology of $\mathcal{O}_p$, in \cref{Cor energy spectral seq} (see also \cref{Prop appendix spectral sequence and E1 page}), after taking those same (co)limits.
From now on, we label the components $M_i$ of $B_p$ by $B_{p,\beta}$. Ordinary cohomology means with coefficients in $\k$, e.g.\,$H^*(Y):=H^*(Y,\k)$ (see \cref{Rmk about coeffs Novikov}). We deduce:

\begin{cor}\label{CorSpectralSeqForSH}
There is a convergent spectral sequence of $\k$-modules,
E_{r}^{pq}(\Fi;\mathcal{P}) \Rightarrow SH_{\mathcal{P}}^*(Y,\Fi), \textrm{   where } E_{1}^{pq}(\Fi;\mathcal{P})=  \begin{cases*}
H^q(Y) & if $p=0, \textrm{ and } 0 \in \mathcal{P}$\\
\bigoplus_{\c} H_{p,\c}^*[-\mu(B_{p,\c})] & if $p<0, \textrm{ and } T_p \in \mathcal{P}$   \\
0 & otherwise,
\end{cases*}
Taking $\mathcal{P}=(0,\infty]$ and $\mathcal{P}=[0,\infty]$,  defines two convergent spectral sequences:
\begin{equation} E_+(\Fi)_r^{pq} \Rightarrow SH_+^*(Y,\Fi), \textrm{   where } E(\Fi)_1^{pq}=  \begin{cases*}
\bigoplus_{\c} H_{p,\c}^*[-\mu(B_{p,\c})] & if $p<0,$    \\
0 & otherwise,
\end{cases*}
\end{equation}
\begin{equation}
E(\Fi)_r^{pq} \Rightarrow SH^*(Y,\Fi), \textrm{   where } E(\Fi)_1^{pq}=  \begin{cases*}
H^q(Y) & if $p=0,$\\
\bigoplus_{\c} H_{p,\c}^*[-\mu(B_{p,\c})] & if $p<0,$    \\
0 & otherwise.
\end{cases*}
\end{equation}
\end{cor}
When $c_1(Y)=0$, as $SH^*(Y,\Fi)=0$ by \cref{Prop vanishing of SH}, the spectral sequence $E(\Fi)^{p,q}_r$
in fact converges to zero; also, Proposition \ref{SpSeqIs1periodic} implies that for sufficiently large $r$ we have $E(\Fi)^{p,q}_r=0$ for all $p,q$.

\begin{cor}\label{CorSpectralSeqAgree}
For columns $p$ of slope $T_p\leq \lambda$, $E(\varphi)_r^{pq}=E_r^{pq}(\Fi;[0,\lambda])$, and $$E_r^{pq}(\Fi;[0,\lambda])\Rightarrow SH^*_{[0,\lambda]}(Y;\Fi) \cong HF^*(H_{\lambda}).$$ Moreover, $HF^*(H_{\lambda})\cong HF^*(\lambda H)$, so when $H^*(Y)$ lies in even degrees also $HF^*(H_{\lambda})$ is supported in even degrees and agrees with the right-hand side of \cref{pureHam}.
\end{cor}
\begin{proof}
This follows by Corollary \ref{CorSubcomplexTrick}, and the fact that Floer cohomology only depends on the choice of slope at infinity for the Hamiltonian, not on the specific construction of the Hamiltonian.
\end{proof}

\begin{rmk}[{\bf Conventions about the spectral sequence Figures.}]\label{Remark Conventions about the spectral sequence Figures}
To illustrate that Corollary, the figures in \cref{ExamplesSpSeq} (which involve $Y$ with $c_1(Y)=0$) have arrows to indicate the edge homomorphisms from the $E_1$ page onwards that are necessary for the spectral sequence to converge to $SH^*(Y,\Fi)=0$ (each dot in the figures indicates a rank one summand). We call these the {\bf iterated differentials}. They need to be suitably interpreted: one must take cohomology iteratively, by first considering arrows that only move one column to the left to determine the cohomology $E_2$, then the arrows that move two columns to the left define a differential on the resulting $E_2$ and it computes $E_3$, etc.\;(cf.\;the proof of \cref{PropTwoRowSplitSpectralSeq}).
We abusively place $H^*(B_{p,\c})[-\mu(B_{p,\c})]$ in the $E_1$-columns rather than the correct $H_{p,\c}^*[-\mu(B_{p,\c})]$. Thus, within each column, there is a further energy-spectral sequence from $H^*(B_{p,\c})[-\mu(B_{p,\c})]$ that converges to $H_{p,\c}^*[-\mu(B_{p,\c})]$. Loosely, it means there may be {\bf vertical differentials} (of degree $1$) within the same column that have not yet been taken into account, which are not indicated in the Figures. This explains why certain boxes (such as the arrow from row 1 to row 2 in \cref{S32_sp_seq}) seem to allow vertical differentials. Vertical differentials can often be excluded a posteriori when all classes are needed to kill $H^*(Y)$, or by using \cref{Lemma excluding vertical drops between certain Bott manifolds} or \cref{Lemma small energy cascades near MB mfd}. However, for $\mathcal{S}_{32}$ in \cref{S32_sp_seq}, in the $B_{1/3}$ column we have seven components $B_{1/3,\beta}$ whose cohomology is that of: $S^3$, two $S^2\times S^1$, four $S^1$. Here we cannot exclude a vertical differential from (one or two) top degree classes of the $S^1$'s, which might kill middle-top degree classes of the $S^2\times S^1$'s.
There can also be vertical differentials in the local Floer cohomology of some $S^2\times S^1$, whereas there cannot be a vertical differential for the four $S^1$ because $\omega$ is exact on $S^1$ (using \cref{Lemma small energy cascades near MB mfd}).
\end{rmk}

Finally, we can relate the spectral sequence $\Epqr$ with
the filtration $\FF^{\varphi}_\lambda$ from \cref{FiltrationOnSingCohomology}.
\begin{prop}\label{FiltrationOnCohomologyBySpectralSequence} $\FF^{\varphi}_\lambda $ is the subspace of the 0-th column $E(\Fi)_1^{0*}\cong H^*(Y)$ that gets killed by images of iterated
differentials for $E(\Fi)_r^{pq}$ defined on columns $p$ of slope $T_p\leq \lambda.$
\end{prop}
\begin{proof}
$\FF^{\varphi}_\lambda $ is defined as the kernel of the continuation map $HF^*(H_{\lambda_0}) \fun HF^*(H_\lambda)$ (where $\lambda_0>0$ is a small slope), and we ensured that at the chain level $CF^*(H_{\lambda_0}) \fun CF^*(H_\lambda)$ is an inclusion of subcomplexes, so the cycles in $CF^*(H_{\lambda_0})$ that get iteratively killed precisely yield that kernel.
\end{proof}
\section{Properties of the Morse--Bott--Floer spectral sequences}\label{SpecSeqMBF-Properties}

\subsection{Conventions}\label{NotationForSpSeq}

We continue with the initial assumption from \cref{filtrationFloer}, and that $c_1(Y)=0$.

We will work with $\Epqr,$ but analogous results hold for $E_+(\Fi)^{p,q}_r$ and $E_r^{p,q}(\Fi;\mathcal{P})$ after specifying cases about $p$, e.g.\,whether or not $T_p$ lies in the period-bracket $\mathcal{P}$.

Ordinary cohomology is computed over $\k$ coefficients (see \cref{Remark choice of coefficients}), so $H^*(Y):=H^*(Y,\k)$.

\textbf{Morse--Bott submanifolds} will refer to the connected manifolds $B_{p,\c}$ of 1-orbits of $H_\lambda$.
Recall by \cref{Cor dim of Bpbeta} that
these are precisely the smooth submanifolds $$B_{p,\c}=\{c'(H)=T_p=\tfrac{k}{m}\} \cap Y_{m,\c}$$ obtained by slicing the non-compact $m$-torsion submanifolds,
where $k,m$ are coprime positive integers.

The column $p$ of the spectral sequence is called \textbf{the time-$T_p$ column}.
For example, the time-$1$ column of the $E_1$-page is the degree-shifted cohomology $H^*(\Sigma)$ of the whole slice $\Sigma:=\{c'(H)=1\}$.

We \textbf{number the rows of the spectral sequence by the total degree} $p+q,$ rather than by $q$.

The gradings will be written in terms of the function $\W(x)$, from \cref{AppendixFloer}.

It is helpful to read the following results while comparing them with the examples in \cref{ExamplesSpSeq}, keeping in mind the conventions explained in \cref{Remark Conventions about the spectral sequence Figures}.

\subsection{Grading of the Morse--Bott manifolds of $1$-orbits $B_{p,\c}$}

\begin{prop}\label{CZindecesOfMorseBottSubmanifolds} The grading of a Morse--Bott submanifold $B_{p,\c}$ is an even integer equal to
\begin{equation}\label{CZindecesOfMorseBottSumbanifolds}\textstyle
\begin{split}
\mu(B_{p,\c}) & = \textstyle \dim_{\C} Y - \frac{1}{2}\dim_{\R} B_{p,\c}-\frac{1}{2}- \sum_i \W(T_p w_i)\\ & \textstyle = \codim_{\C}\, Y_{m,\c} - \sum_i \W(T_p w_i),
\end{split}
\end{equation}
where $w_i$ are the weights
of the convergence point $\x_{\iinfty}$ of an arbitrary point $\x\in B_{p,\c}.$

For the hypersurface $\Sigma=\{ c'(H)=1\}$ corresponding to the period $T_p=1$:
\begin{equation}\label{Equation shift down for integer time hypersurfaces}
\qquad  \qquad  \qquad  \qquad  \qquad \mu(\Sigma)=-2\mu, \qquad \textrm{and }\qquad \mu(\{ T_p=N\})=-2N\mu \quad\textrm{ for }N\in \N,
\end{equation}
where $\mu=\sum w_i$ is the Maslov index of the $S^1$-action (\cref{Subsection discussion of ma and muFalapha}).
\end{prop}
\begin{proof} We are refining the computation from \cref{Thm Floer traj near Bi new perspective}, and the proof idea is very similar to \cite[Sec.3.3]{OanceaEnsaios}. In short: the difference between the Hamiltonian flows of $H_\lambda$ and $T_p H$ causes a symplectic shear which gives the $-\frac{1}{2}$ term in \eqref{CZindecesOfMorseBottSumbanifolds}, and the Robbin-Salamon index of $B_{p,\c}$ with respect to the Hamiltonian $T_p H$ will yield the $\sum_i \W(T_p w_i)$-part like in the proof of \cref{PropSec2RSIndices} in [41].

For an arbitrary point $y\in B_{p,\c},$ %
recall  the orthogonal splittings $T_y Y= \la X_{\R_+} \ra \oplus \la X_{H_\lambda} \ra \oplus \xi$ and $T_\x \Sp =X_{H_\lambda}\oplus \xi$ from \eqref{splittingTangentSpaceContact}. %
Using this splitting, define the matrices
\[
\begin{bmatrix}
{\tau c''(H) \norm{\nabla H}^2}&1&\\
&& {(\phi_t^{H_\lambda})_*|_\xi}
\end{bmatrix},
\qquad \qquad
\chi(t)=
\begin{bmatrix}
{\tau c''(H) \norm{\nabla H}^2}&1&\\
&& Id_{2d-2}
\end{bmatrix}.
\]
We claim that $F_t$ is the matrix for the linearised flow $(\phi_t^{H_\lambda})_*$ of $H_\lambda$. Indeed the first column is due to Lemma \ref{LemmaForTheShear}; the second column is immediate; and the rest is due to the fact that the flow $\phi_t^{H_\lambda}$ restricted to $\Sp$ is equal to $\phi_t^{T_p H},$ hence preserves the metric and thus the orthogonals.

As $\phi_t^{H_\lambda}$  preserves $X_{H_\lambda},$ it preserves its orthogonal $\xi.$
Thus, we have that $(\phi_t^{H_\lambda})_*=  \chi(t) \circ (\phi_t^{T_p H})_*.$
Letting $\Psi(t)=(\phi_t^{T_p H})_*$ as in the proof of \cite[Prop.2.2]{CFHW}, there is a homotopy

$$K(s, \, t) = \left \{ \begin{array}{ll} \chi(st) \Psi(\frac {2t} {s+1})
, & t \le \frac {s+1} 2 \\
\chi\big( (s+2)t - (s+1) \big) \Psi(1) , & \frac{s+1} 2 \le t
\end{array} \right.
of paths with fixed ends between the path $(\phi_t^{H_\lambda})_*$ and the concatenation of
$(\phi_t^{T_p H})_*$ and $\chi(t) \circ (\phi_1^{T_p H})_*.$

Hence, by Theorem \ref{RSProperties}.(\ref{catenation}),(\ref{shear}) we have $RS(x,H_\lambda)= RS(x, T_p H) +
\frac{1}{2},$ where $x=x(t)$ denotes
the 1-orbit of $H_\lambda$ and $T_p H,$ starting at $y=x(0).$
Thus, it remains to compute the index $RS(x,T_p H).$
We have to pick a symplectic trivialisation of $x^*TY$ first.
By Lemma \ref{LemmaCanonicalFillingDiscs},
a choice of unitary basis $v_i$ of tangent vectors at $y_0$ with weights $w_i$ yields a symplectic trivialisation $v_i(z)$ of $\psi_y^*TY$, where
$\psi_y$ is the canonical holomorphic capping disc from \cref{LemmaCanonicalFillingDiscs}.
As $T_p=k/m$ for coprime $k,m$, the $S^1$-orbit $S^1\cdot y$ is an $m$-fold cover of a closed $S^1$-orbit $\hat{y}$ of minimal period $1/m$, and $v_i(z)$ is $\Z/m$-equivariant by \cref{LemmaCanonicalFillingDiscs}. In particular $v_i(z)$ determines a unitary trivialisation of $\hat{y}^*TY$. As $T_p=k/m$, $x(t)$ is a $k$-fold cover of $\hat{y}$, so $v_i(z)$ determines a unitary trivialisation of $x^*TY$.
Thus we obtain a canonical unitary and symplectic trivialisation for $x^*TY$ determined by the choice of $v_i$.

By continuity of Robbin-Salamon indices, Theorem \ref{RSProperties}, $RS(x,T_pH)=RS(y_0,T_pH)$ where $y_0=\lim_{t\to 0}\varphi_t(y)$ is the convergence point, viewing $y_0$ as a constant $1$-orbit for the Hamiltonian $T_p H$.
On the weight decomposition $T_{y_0} Y = \oplus \C_{w_i}$ the linearisation of the flow of $T_p H$ is rotation on $\C_{w_i}\cong \C$ with speed $T_p w_i$, which contributes $\W(T_p w_i)$ to the RS-index by \cref{RSProperties}. In summary:
RS(x,H_\lambda)= RS(x, T_p H) +
\frac{1}{2} \qquad\textrm{ and }\qquad RS(x,T_pH) = RS(y_0,T_p H)=\sum_i  \W(T_p w_i).

The grading of the Morse-Bott manifold $B_{p,\c}$ corresponds to the grading of the orbit that represents the minimum of the auxiliary Morse function $f_{p,\c}:B_{p,\c}\to \R.$
By the argument in \cite[Sec.3.3]{OanceaEnsaios}, the Morse term in the Robbin-Salamon index formula for an orbit is equal to half of the signature of the Hessian $\nabla f_{p,\c}$ at the corresponding critical point; thus at a minimum it is equal to $\frac{1}{2}\dim_{\R} B_{p,\c}.$ By our grading conventions from \cref{GradingConventions}, the formula \eqref{CZindecesOfMorseBottSumbanifolds} follows.

To prove that $\mu(B_{p,\c})$ is even, it suffices to show that $\sum_i \W(T_p w_i) \equiv \codim_{\C}\, Y_{m,\c} \;(\textrm{mod 2})$.
Recall $T_p= k/m$, for coprime $k,m$, so
$T_p w_i\in  \Z$ precisely when $m|w_i.$ %
By \cref{Lemma torsion submanifolds}, $T_{y_0}Y_{m,\c}=\oplus H_{ma}\subset T_{y_0}Y$
is summing over the same set of weights.
Finally, by \eqref{WfunctionForRSAppendix}, $\W(T_p w_i)$ is even precisely when $T_p w_i\in  \Z,$ or equivalently when $m|w_i,$ %
so $\sum \W(T_p w_i)$ modulo $2$ is the complex dimension of the vector space complement of $T_{y_0}Y_{m,\c}\subset T_{y_0}Y$.

Using \eqref{WfunctionForRSAppendix}, %
$\sum  \W(2\pi w_i)=\sum 2 w_i=%
2\mu$ and $\tfrac{1}{2}(\dim_{\R} \Sigma + 1) = \dim_{\C} Y$, and the final claim follows.
\end{proof}

\begin{cor}\label{Rmk index of Bott manifolds in terms of muFa}
Let $T_p= \tfrac{k}{m}$ with $k,m$ coprime, and $T_p^{\pm}$ non-critical times just above/below $T_p$. Let $\F_\a$ be the minimal component of $\Ymc$, and $\mathrm{rk}(Y_{m,\c})$ the rank from \cref{Remark local rank}. Then
$$\mu(B_{p,\c})= \mu_{T_p}(\F_\a) - \mathrm{rk}(Y_{m,\c})=\mu_{T_p^+}(\F_\a),$$
which is the lowest degree in which $H^*(B_{p,\c})[-\mu(B_{p,\c})]$ is supported, and the top degree is:
\mu(B_{p,\c})+\dim_{\R}B_{p,\c} =
\mu_{T_p}(\F_\a) + \dim_{\C} Y_{m,\c} + \dim_{\C} \F_\a -1 = \mu_{T_p^-}(\F_\a) + 2\dim_{\C} \F_\a - 1.
For $T_p<1$ it is supported in degrees $\geq 2\dim_{\C}Y-2\mu-t$.

If $Y$ is compatibly-weighted (\cref{Definition compatibly weighted}, e.g.\,all CSRs), then
$H^*(B_{p,\c})[-\mu(B_{p,\c})]$ is supported strictly below degree $t:=\max \{n\in \N: H^n(Y)\neq 0\}$.

For a CSR of weight $s=1$, for $T_p<1$ it is supported in $[0,t-1]$, and for $s=2$ in $[-t,t-1]$.
\end{cor}
\begin{proof}
Abbreviate $|V|:=\dim_{\C} V$.
In each case, the first equality follows by \cref{Lemma muFa is smaller than MB index mua} and $\dim_{\R}B_{p,\c}=2|Y_{m,\c}|-1$,
the second equality uses \cref{Lemma Fma is jump in grading}:
$\F_\a$ is $m$-minimal, so
$F_m^\a = \mathrm{rk}_y(Y_{m,\c}) = |Y_{m,\c}|-|\F_\a|$, and we have $\mu_{T_p^+}(\F_\a)=\mu_{T_p}(\F_\a)- \mathrm{rk}(Y_{m,\c})$ and
$\mu_{T_p^-}(\F_\a) = \mu_{T_p}(\F_\a)+F_m^\a$. So
\mu_{T_p}(\F_\a) + |Y_{m,\c}| + |\F_\a| -1
= \mu_{T_p^-}(\F_\a) + 2|\F_\a| - 1.
When $Y$ is compatibly-weighted, $\mu_{T_p^-}(\F_\a)
\leq \mu_\a$ by
\cref{Lemma muFa is smaller than MB index mua},
and $\mu_\a + 2|\F_\a|\leq t$ by \cref{EqnFrankel}, so $\mu_{T_p}(\F_\a) + |Y_{m,\c}| + |\F_\a| -1\leq t-1.$
For the claim about $T_p<1$, we use \cref{CentralSymmetrySpecSeq}: $t-1$ is reflected to $2(|Y|-\tfrac{1}{2}-\mu)-(t-1)=2|Y|-2\mu-t$. For CSRs we have $2\mu=s\cdot |Y|$ and $t\leq |Y|$, with $t=|Y|$ when $s=1$ (using \cref{CohomologyOfACSRProperties}).
\end{proof}

\subsection{The columns of the $E_1$-page are 1-periodic up to downward $2\mu$-shifts}

\begin{cor}\label{CorMuForTauTau'}
If $T_{p}=T_{p'}+N,$ for $N\in \N$, then
\end{cor}
\begin{proof}
We use  %
\cref{LemmaSec2MaslovIndex} ($\mu=\sum w_i$),
\cref{CZindecesOfMorseBottSumbanifolds}, and the property $\W(x+N)=\W(x)+2N$:
\mu(B_{p',\c})\!-\!\mu(B_{p,\c})=\textstyle \sum [\W(T_{p'} w_i)-\W(T_p w_i)] %
=\sum [\W((T_{p}+N) w_i)-\W(T_p w_i)]= \sum 2Nw_i=2N\mu. \qedhere

\end{proof}

\begin{cor}(Periodicity)\label{SpSeqIs1periodic}
The $E_1$-page of the spectral sequence $\Epqr$ is time-1 periodic up to shifting the rows downwards by $2\mu$ rows.
For a weight-$s$ CSR, $2\mu = s \cdot \dim_\C Y$ by \cref{GradientIsRPlus}.
\end{cor}
Thus, in order to compute the $E_1$-page of $\Epqr,$ it is enough to compute the columns $p$ with  period $T_p\leq 1$. We show now that it suffices to consider $T_p \leq \frac{1}{2}$ due to a local central symmetry.

\subsection{Local central symmetry in the $E_1$-page}

In Figure \ref{SS_S41} the 4-star symbol lies at the point ``$(\frac{1}{2},-\frac{1}{2})$''. In general, the \textbf{point ``$(\frac{1}{2},n-\frac{1}{2})$''} in the table for the spectral sequence lies on the column with time $T_p=\frac{1}{2},$ and on the line between rows $n$ and $n-1.$ If a column with $T_p=\frac{1}{2}$ does not exist, then it lies on the line between the two columns whose periods are closest to $\frac{1}{2}$.

\begin{cor}[Local central symmetry] \label{CentralSymmetrySpecSeq}
There is a central symmetry of the sub-block of the $E_1$-page of $\Epqr$ consisting of columns $p$ with $0<T_p<1,$ about the point
$$\textstyle \left(\frac{1}{2}, \dim_\C Y - \frac{1}{2} - \mu  \right).$$
If $Y$ is a weight-1 CSR it is the point $(\frac{1}{2}, \frac{1}{2}\dim_\C Y -\frac{1}{2})$, for a weight-2 CSR it is $(\frac{1}{2},-\frac{1}{2})$.
\end{cor}
\begin{proof}
By (\ref{CZindecesOfMorseBottSumbanifolds}), the top-dimensional class $H^{top}(B_{p,\c})$ lies in the row
$$\textstyle r_1:=\dim_{\R} B_{p,\c}+\dim_{\C} Y - \frac{1}{2}\dim_{\R} B_{p,\c}-\frac{1}{2}- \sum_i \W(T_p w_i).$$ For $T_p<1$, in the column $p'$ such that $T_{p'}=1-T_p<1$, there is a Morse-Bott manifold $B_{p',\c}$ diffeomorphic to $B_{p,\c}$ by Lemma \ref{Only the primitive Ones matter}.
By (\ref{CZindecesOfMorseBottSumbanifolds}), its bottom-dimensional class $H^{0}(B_{p',c})$ lies in the row
$$\textstyle r_2:=\dim_{\C} Y - \frac{1}{2}\dim_{\R} B_{p',\c}-\frac{1}{2}- \sum_i \W(T_p' w_i).$$
Thus $\frac{r_1+r_2}{2}=\frac{1}{2}(2 \dim_\C Y -1 - \sum_i (\W(T_p w_i)+\W(T_{p'} w_i))=\frac{1}{2}(2 \dim_\C Y -1- \sum_i 2w_i)=\dim_\C Y - \frac{1}{2} - \mu,$ where
we used $\W(-a)=-\W(a)$ and $\W(N +a)=2N+\W(a)$.
So $H^{top}(B_{p,\c})$ and $H^{0}(B_{p',\c})$ match-up via the
central symmetry about the point $(\frac{1}{2},\dim_\C Y - \frac{1}{2}- \mu ).$ In the other degrees, the ranks match-up due to Poincaré duality and the fact that $B_{p,\c}\iso B_{p',\c}.$
\end{proof}
\begin{cor}[Symmetries revisited] \label{CentralSymmetrySpecSeq2}
The periodicity in \cref{SpSeqIs1periodic} and local central symmetry in \cref{CentralSymmetrySpecSeq} also hold for the groups $H_{p,\c}^*[-\mu(B_{p,\c})]$ from \cref{CorSpectralSeqForSH}.
\end{cor}
\begin{proof}
Let $\Sigma_p:=\{H=H_p\}=\{c'(H)=T_p\}$.
The periodicity is a consequence of the method from \cref{Subsection A comment about Hamiltonian perturbations} for $G:=c(H)$, $P:=H$, $p:=S^1$-flow (the flow for $H$). We get $p^*I=I$ (as the $S^1$-flow is pseudoholomorphic) and $p^*G=\widetilde{c}(H)-H$, and a bijection of the local Floer complex for $(\widetilde{c}(H)-H,I)$  to the local Floer complex for $(\widetilde{c}(H),I)$, via $x\mapsto p\circ x$ that decreases the grading by $2\mu.$ More precisely, we work with {\MB} Floer complexes (see \cref{AppendixCascades}), we use the same auxiliary {\MB} functions for $B_{T_p,\beta}$ and $B_{1+T_p,\beta}$, and the above bijection ensures the required bijections of the drops in the cascades. So the generators and {\MB} Floer differentials are identified for the local {\MB} Floer complexes for $\Sigma_p$ and $\Sigma_{p'}$ where $T_{p'}=1+T_p$.

We now compare the local Floer cohomologies of a neighbourhood $N$ of $c'(H_p)=T_p$ and a neighbourhood $N'$ of $c'(H_{p'})=T_{p'}:=1-T_p$.
Using the flow $\psi_t$ of $\tfrac{\nabla H}{\|\nabla H\|^2}$ we can identify $N \to N'$.
The flow yields a family $\psi_t^*I,\psi_t^*\omega$ of data on $N$, telling us how to deform the data $I,\omega$ so that local Floer cohomology of $N$ can be identified with that of $N'$ if we were using the same Hamiltonian under the identification $N\to N'$ (which we are not). On $N$ we are using the Hamiltonian $c(H)$, and using a {\MBF} model involving auxiliary Morse functions $f_{\beta}:B_{p,\beta}\to \R$ (\cref{AppendixCascades}).
The local {\MB} Floer cohomology of $(N,c(H))$ is invariant up to isomorphism under changing the precise choice of $c$ provided that we keep, say, our {\MB} manifolds of $1$-orbits fixed and ensure the slope $c'(H_p)=T_p$ and $c''(H_p)>0$ there, call this ``$T_p$-data''.
If we instead have $c''(H_p)<0$, call it ``$T_p$-antidata''.
Observe that the local {\MB} Floer cohomology for $T_p$-data $c(H)$ and for $T_p$-antidata $\widetilde{c}(H)$ are isomorphic\footnote{This is because if we locally concatenate $c(H)$ with $\widetilde{c}(H)$, one can homotope the concatenation to a Hamiltonian with slope less than $T_p$ whose local Floer cohomology therefore vanishes. Thus the (degree one) {\MB} Floer differential from the complex for the $T_p$-data must map quasi-isomorphically to the complex for the $T_p$-antidata.} via a grading shift $d\mapsto d+1$. This shift by $+1$ can also be explained in terms of the shear in \eqref{Grading equation in appendix} (see also the proof of \cref{CZindecesOfMorseBottSubmanifolds}): for the $T_p$-data we get a contribution of $-\tfrac{1}{2}$ to the grading, whereas for the $T_p$-antidata we get $+\tfrac{1}{2}.$

It remains to compare slope $(1-T_p)$-data on $N$ and slope $T_p$-data on $N$. By the shifting argument at the beginnning of the proof, slope $(1-T_p)$-data has the same local Floer cohomology as $-T_p$-data after changing degrees by $d\to d+2\mu$.
Suppose $\gamma(H)$ is the $-T_p$-data obtained.

We have a canonical duality isomorphism
of local {\MB} Floer complexes:
\cong
which sends $x\in \mathrm{Crit}(f_{\beta})$ to $x\in \mathrm{Crit}(-f_{\beta})$, and the {\MB} Floer differentials coincide.\footnote{We reverse the flow-direction of ledges, and we reverse both coordinates for drops, $u(s,t)\to u(-s,-t)$.}
Note that $-\gamma(H)$ is $-T_p$-antidata, since $-\gamma''(H_p)<0$. As explained previously, we can identify this with the local {\MB} Floer cohomology of $T_p$-data after index shifting $d\to d-1.$

In conclusion, going from $(1-T_p)$-data to $T_p$-data arose from quasi-isomorphisms at local {\MB} Floer chain level subject to shifting degrees by
$$d\to d+2\mu \to 2\,\dim_{\C}Y-(d+2\mu) \to 2\,\dim_{\C}Y-(d+2\mu)-1 = 2\,\dim_{\C}Y-d-2\mu-1,$$
which is a reflection of $d$ about $\dim_{\C} Y - \tfrac{1}{2}-\mu$ of $d$ in $\R$, as required in \cref{CentralSymmetrySpecSeq}.
\end{proof}

\subsection{Bounds on where the $E_1$-page is supported}

\begin{de}
The {\bf block number} is the integer part $N:=\lfloor T_p \rfloor$ of the period.
{\bf Pillars}
are columns of $E(\Fi)_1^{pq}$ for integer periods $T_p$: the $0$-pillar is $H^*(Y)$, the $N$-pillar is $H^*(\Sigma)[2N\mu]$ for $N\geq 1$.

The {\bf $N$-block} is the sub-block of $E(\Fi)_1^{pq}$ of classes with block number $N$ excluding the pillars, so $N<T_p<N+1$.
A pillar/block is {\bf supported} in $[a,b]$ if it vanishes for grading values outside of $[a,b]$.
\end{de}

\begin{lm}\label{Lemma block description spectral sequence}
Suppose all $\mu_{\lambda}(\F_\a)\leq \mu_\a$ %
(e.g.\,any compatibly-weighted $Y$, \cref{Definition compatibly weighted}, e.g.\,CSRs).

Let $y:=\dim_{\C} Y$ and $t:=\max \{n\in \N: H^n(Y)\neq 0\}.$
Then $t\leq 2y-2$ is even, and we have:
\begin{center}
\begin{tabular}{c|c|c}%
\strut & lower bound on support & upper bound on support \\%& comments \\
\hline
$0$-pillar & $0$ & $t$ \\%& both bounds are achieved\\
$0$-block & $2y-2\mu-t$ & $t-1$ \\%& upper bound $<$ top of $0$-pillar\\
$1$-pillar & $-2\mu$ & $2y-1-2\mu$ \\%& \strut \\%both bounds are achieved: DON'T KNOW DUE TO VERTICAL DIFFERENTIALS
$1$-block & $2y-4\mu-t$ & $t-1-2\mu$ \\%&  upper bound $<$ top of $1$-pillar: DON'T KNOW DUE TO VERTICAL DIFFERENTIALS\\
$N$-pillar & $-2N\mu$ & $2y-1-2N\mu$ \\%& \strut \\%both bounds are achieved\\
$N$-block & $2y-2(N+1)\mu-t$ & $t-1-2N\mu$ \\%&  upper bound $<$ top of $k$-pillar
\end{tabular}
\end{center}
The $0$-block (resp.$1$-pillar) after adding $-2N\mu$ to the grading is the $N$-block (resp.$(N+1)$-pillar).\\
The blocks have a local central symmetry analogous to the $0$-block (\cref{CentralSymmetrySpecSeq}).

For a weight-$s$ CSR we have $2\mu=sy$ and $t\leq y$ (with $t=y$ for $s=1$), so:\\ %
For $s=2$: the $0$-block is supported in $[-t,t-1]$; the $1$-pillar in $[-2y,-1]$.\\
For $s=1$: $0$-block $\subset [0,y-1]$;  $1$-pillar $\subset [-y,y-1]$; $1$-block $\subset [-y,-1]$; $2$-pillar $\subset [-2y,-1]$.
\end{lm}
\begin{proof}
In \cref{EqnFrankel}, $\F_\a$ is a closed $I$-pseudoholomorphic  submanifold, so it is even-dimensional, and $\mu_\a$ is even, so $t$ is even (and $t\neq 2y$ as $Y$ is non-compact), so $t\leq 2y-2$.
For $N\geq 1$, the $N$-th pillar arises from an energy-spectral sequence starting with
$H^*(\Sigma)[2N\mu]$, so is supported in $[-2N\mu,-2N\mu+\dim_{\R}\Sigma]$, and $\dim_{\R}\Sigma = 2y-1$.
The rest follows by \cref{Rmk index of Bott manifolds in terms of muFa} and \cref{SpSeqIs1periodic}.
\end{proof}

\subsection{What classes in the $E_{\geq 1}$-pages can hit $H^*(Y)$}

By \cref{SpSeqIs1periodic} and \cref{FiltrationOnCohomologyBySpectralSequence}, for the purposes of computing the filtration $F^{\varphi}_\lambda$ we only need to consider columns up to a certain time: %

\begin{cor}\label{HitingTimeOfColumns}
Suppose all $\mu_{\lambda}(\F_\a)\leq \mu_\a$ %
(e.g.\,any compatibly-weighted $Y$, \cref{Definition compatibly weighted}, e.g.\,CSRs).

The only $N$-blocks that can potentially hit the 0-pillar via iterated differentials satisfy  $N\leq t/2\mu$.
The only $N$-pillars that can potentially hit the 0-pillar via iterated differentials satisfy  $N\leq y/\mu$.
For a weight-$s$ CSR those two conditions become
$N \leq t/sy\leq 1/s$ and $N\leq 2/s$
\end{cor}
\begin{proof}
This will follow by \cref{Lemma block description spectral sequence}. For the $N$-block, the highest possible grading $t-1-2N\mu$ is strictly less than $-1$ if $N>t/2\mu$, so the iterated differentials (which increase the total grading by $1$) from such an $N$-block cannot hit the $0$-pillar. Higher numbered blocks lie in lower gradings, so the same argument applies. Similarly, for the $N$-pillar we have $2y-1-2N\mu< -1$ if $N>y/\mu$.
\end{proof}
When $H^*(Y)$ lies in even degrees,  the only non-zero iterated differentials go from odd numbered rows to even rows (as illustrated by the figures in \cref{ExamplesSpSeq}):

\begin{prop}\label{PropTwoRowSplitSpectralSeq}
Suppose $H^*(Y$) lies in even degrees ($\Leftrightarrow$ all $H^*(\F_\a)$ lie in even degrees), e.g.\,any CSR.
Then from the $E_1$-page onwards, the spectral sequence $\Epqr$ splits horizontally into a direct sum of two-row spectral sequences. Also, on any given odd degree class $x$ of the $E_1$-page, some iterated differential is eventually injective on $x$, so kills some class in a column strictly to the left of $x$.
\end{prop}
\begin{proof}
In this proof, write $E_r(H_{\lambda})$ to mean the $E_r$-page for the spectral sequence for $H_{\lambda}$.
By induction on the number of the column to the right of the $0$-pillar, we will prove that: all iterated differentials $d_j$ vanish on even classes, and
for any odd degree class $x$ we have $d_1(x)=0,$ $d_2(x)=0$, $\ldots$, $d_{r-1}(x)=0$, $d_r(x)\neq 0$ for some $r\geq 1$ possibly equal to $1$, in particular $x$ kills some
non-trivial class in a column strictly to the left of $x$ via an iterated differential.
The initial step of the induction holds since
the $0$-pillar is supported in even degrees, and columns to the left of the $0$-pillar are zero.
Suppose that
the induction holds up to columns with $T_p< b$,
and consider $b<\lambda$ with %
$E_1(H_{b})$ and $E_1(H_{\lambda})$ differing in just one column, so a column arising for time $b<T_p<\lambda$. We call ``new'' that column and its classes, and ``old'' the columns to the left and their classes.
If there is an odd new class $x\neq 0$ with $d_j(x)=0$ for all $j\geq 1$,
then $[x]$ would be a non-trivial class in all $E_j(H_{\lambda})$ (as all columns in $E_1(H_{\lambda})$ to the right of the new column are zero, so $x$ is not in the image of any $d_j$). This would yield a non-trivial odd class in  $HF^*(H_{\lambda})$, contradicting
\cref{pureHam}.
Now let $y$ be an even new class; we want to show $d_j(y)=0$ for all $j\geq 1$.
By contradiction, consider the smallest $j\geq 1$ for which that fails. Then $d_j(y)=x$ kills an odd old class $[x]$ that survived to $E_j(H_{b})$.
By the inductive hypothesis for $H_{b}$, $d_{j-1}(x)=\cdots = d_{r-1}(x)=0$, $d_r(x)\neq 0$ (with $r\geq j$ possibly equal to $j$).
The iterated differentials of $x$ for $H_{b}$ agree with those for $H_{\lambda}$, indeed $(E_i(H_{b}),d_i)\subset (E_i(H_{\lambda}),d_i)$ is a subcomplex.\footnote{by the $F$-filtration, for $x\in E_*(H_{b})$, $d_i(x)$ only involves Floer trajectories trapped in the region where $H_{\lambda}=H_{b}$.}
The exact class
$d_j(y)\in E_r(H_{\lambda})$ is zero so $d_r(d_j(y))=d_r(x)\in E_r(H_{\lambda})$ must be zero. As $d_r(x)$ is not zero in $E_r(H_{b})$, there is a new class $w$ which survives to $E_r(H_{\lambda})$ with $d_r(w)=d_r(x)$. This is impossible: $d_r(w)$ cannot reach the column where $d_r(x)$ lives ($w$ lies in a column strictly to the right of $x$).
\end{proof}

\subsection{Computing the $E_1$-columns arising from hypersurface sections of vector bundles}
\label{Subsection Techniques for computing the $E_1$-page}

\begin{lm}\label{CohomologicalIndependenceOnHypersurface}
Let $M$ be an open manifold, and $H: M\to \R$ an exhausting function whose critical locus is compact.
Let $\Sigma_c=H^{-1}(c)$ denote the level set.
Then, for sufficiently large $c\in \R$, the cohomology $H^*(\Sigma_c)$ is independent of $c$ and the choice of $H$. In particular, if $M$ is a vector bundle and $S(M)$ a sphere subbundle, $H^*(\Sigma_c)\cong H^*(S(M))$ for large $c$.
\end{lm}
\begin{proof}
For large $c$, the critical locus of $H$ is contained in $H<c$ (as $H$ is exhausting).
Thus, for large $c,c'$ we have a diffeomorphism $\Sigma_c\cong \Sigma_{c'}$ defined by the flow of $\nabla H/\|\nabla H\|^2$.
Denote by $\Sigma_{c,\infty}=H^{-1}[c,\infty)\subset M$ the subset at infinity diffeomorphic to $\Sigma_c\times [0,\infty)$ obtained by flowing for positive time with  $\nabla H/\|\nabla H\|^2$. Thus $H^*(\Sigma_{c,\infty}) \cong H^*(\Sigma_c)$ (for large $c$).
Given two exhausting functions $H'$ and $H''$, respectively with level sets $\Sigma_c'$, $\Sigma_c''$, we can choose $c$-values to get nested sets
\Sigma_{c',\infty}'
\subset
\Sigma_{c'',\infty}''
\subset
\Sigma_{C',\infty}'
\subset
\Sigma_{C'',\infty}'',
where the $c$-values are large so that the critical loci of $H',H''$ lie in lower sublevel sets.
The composite of the first two inclusions is isotopic to a diffeomorphism (by using the flow by $\nabla H'/\|\nabla H'\|^2$). Similarly, for the composite of the last two inclusions (using $H''$). Thus on cohomology those two composite maps induce isomorphisms. Thus $H^*(\Sigma_{c'',\infty}'') \leftarrow H^*(
\Sigma_{C',\infty}')$ is both injective and surjective, so it is an isomorphism. By the first part, up to isomorphism the choices of large $c'',C'$ do not matter.
The second claim follows: let $g$ be the fibrewise metric on $M$ defining $S(M)$, then $H:M\fun \R$, $H(b,\xi)=g(\xi,\xi)$ is an exhausting
function and $S(M)$ is a level set.
\end{proof}
Applying \cref{CohomologicalIndependenceOnHypersurface} to $M=\Ymc$ and $H|_{\Ymc}^{-1}(H_p)=\Sigma_p\cap \Ymc =
B_{p,\c}\subset \Ymc,$ we get:

\begin{enumerate}[(1)]
\item If $Y_{m,\c}=\Hm$ is a torsion bundle (\cref{Defn torsion bundle})
over $\F_\a,$ then
$$H^*(B_{p,\c})\iso H^*(S(\Hm)),$$
which is calculable by considering the Gysin sequence for the sphere bundle $S(\Hm)\to \F_\a$.
\item More generally, if $\mathrm{Core}(\Ymc)=\Ymc\cap \mathrm{Core}(Y)$ is a smooth manifold (not necessarily fixed under the $\C^*$-action), and $\Ymc$ is a smooth vector bundle over it,
$$H^*(B_{p,\c})\iso H^*(S(Y_{m,\c})),$$
which is calculable by considering the Gysin sequence for the sphere bundle $S(\Ymc)\to \mathrm{Core}(\Ymc)$.
\end{enumerate}

\textbf{NB.} In all the examples we consider in \cref{ExamplesSpSeq}, only those two cases occurred for all $B_{p,\c}$.

\subsection{Computing the integer-time columns of the $E_1$-page}

When $H^*(Y)$ is supported in degrees $[0,\dim_{\C}Y]$, we can compute the time-1 column of the $E_1$-page, $H^*(\Sigma)[2\mu]$.
By [41], this assumption %
holds if $\mathrm{Core}(Y)\subset Y$ has the homotopy type of a CW complex of dimension $\leq \dim_{\C}Y$.
This holds for example for all CSRs and
Moduli spaces of Higgs bundles.
\begin{prop} \label{CalculationOfTheSliceCohomology} %

Assume that the Novikov field $\k$ is defined over a base filed $\mathbb{B}$ of characteristic $0$. Let $n=\dim_{\C} Y$, so $\dim_{\R} \Sigma=2n-1$.
Suppose that $H^*(Y)$ is supported in degrees $0\leq * \leq n$. Then
\[H^k(\Spp)\iso  \begin{cases*}
H^{2n-1-k}(Y) & \ $k\geq n+1,$\\
H^{n-1}(Y)\oplus \ker \psi  & \ $k=n,\ n-1,$\\
H^k(Y) & \ $k\leq n-2$.
\end{cases*}
\]
where $\psi: H_{n}(Y) \fun H_n^{lf}(Y)\cong H_{n}(Y,Y_{\infty})$
is the natural map for the pair $(Y,Y_{\infty})$,
and $Y_{\infty}:=H^{-1}[H_p,\infty)$ is the outside of $\Sigma_p=\partial Y_{\infty}$.

In particular, if the intersection form $H_{n}(Y) \times H_{n}(Y) \fun \k$ vanishes, then $\ker \psi \iso H^n(Y)$,
and if it is non-degenerate, then $\ker \psi=0.$
\end{prop}
\begin{proof}
Let $M:=\{H\leq H_p\}$, so $\partial M=\Sigma_p\cong\Sigma$. Thus $Y$ is homotopy equivalent to $M$, as it deformation retracts onto $M$ via the (rescaled) $\R_+$-action.
We have the long exact sequence for the pair $(M,\partial M)$ %
\begin{equation}\label{LESforPair}
\dots \rightarrow H_{k+1}(M,\partial M) %
\rightarrow H_k(\partial M) \rightarrow H_k(M) \rightarrow H_{k}(M, \partial M) \rightarrow \cdots
\end{equation}
By Poincar\'{e}(-Lefschetz) duality
over the field $\k$,  $H_k(M,\partial M)\cong H^{2n-k}(M),$ %
for $k\leq n-2$
the outside terms in \eqref{LESforPair} vanish by the assumption on $H^*(Y)$. Thus $H^k(\partial M)\cong H^k(M)$ (by universal coefficients).  %
By Poincar\'{e} duality for the closed orientable $(2n-1)$-manifold $\partial M=\Sigma$ and universal coefficients, we have $H^k(\Sigma)\cong H_{2n-1-k}(\Sigma)\cong H^{2n-1-k}(\Sigma)$. Thus, for $k\geq n+1$, $H^k(\Sigma) \iso H^{2n-1-k}({Y})$.

Using \eqref{LESforPair} for $k=n$ and $k=n-1$:		%
\begin{equation*}%
H_{n+1}(M)=0\to H_{n+1}(M,\partial M) \fun H_n(\partial M) \rightarrow H_{n}(M) \fja{\psi} H_n(M,\partial M) \rightarrow H_{n-1}(\partial M)\fun H_{n-1}(M).
\end{equation*}
As $H_n(\partial M)/H_{n+1}(M,\partial M)\cong \ker \psi$, the claim about $\psi$ follows by using universal coefficients, and the Poincar\'{e} duality isomorphisms $H_{n+1}(M,\partial M)\cong H^{n-1}(M)$ and
$H^{n-1}(\partial M)\cong H_n(\partial M)$ ($\cong H^{n}(\partial M)$).

The last claim uses the following relation between intersection pairing $\langle \cdot ,\cdot\rangle$ and Poincar\'{e} duality:
\begin{equation}\label{Equation intersection pairing}
\begin{tikzcd}[column sep=small]
\langle \ ,\cdot\rangle:\;H_{n}(M) \arrow{r}{\psi}
& H_n^{lf}(M)\cong H_n(M,\partial M)\;\; \arrow[r,shorten=-2mm,"{\iso}","\textrm{P.D.}"']
& \;\;H^n(M)\arrow{r}{\iso}
& \Hom(H_n(M),\k).
\end{tikzcd} \qedhere
\end{equation}
\end{proof}
\begin{cor}\label{Mid Coh of Hypersurface Vanish}
Let $\M$ be a weight-$s$ CSR, $n:=dim_\C \M$. The cohomology of its hypersurface $\Sigma$ is
\[H^k(\Spp)\iso  \begin{cases*}
H^{2n -1-k}(\M) & \ $k\geq n+1,$\; (in particular vanishes if $k$ is even)\\
0 & \ $k=n-1, n$\\
H^k(\M) & \ $k\leq n-2$, \; (in particular vanishes if $k$ is odd).
\end{cases*}
\]
In terms of the $H_{p,\c}^*[-\mu(B_{p,\c})]$ from \cref{CorSpectralSeqForSH},
$H_{N}^*[-\mu(B_{N})] \cong H^*(\Spp)[Ns\dim_{\C}\M]$ for $N\in \N$.
\end{cor}
\begin{proof}

By \cref{CohomologyOfACSRProperties},
$H^*(\M)$ is supported in even degrees only, with
$0\leq * \leq \dim_{\C} \M$, and its intersection form is non-degenerate due to \cref{CSR_definite_intersection_form}.
Thus \cref{CalculationOfTheSliceCohomology}
The last claim follows because the energy-spectral sequence collapses on the $E_1$-page since there can be no vertical differentials for grading reasons, as there is a $0$-gap between any two degrees in which $H^*(\Sigma)$ is supported.
We also used that $2\mu=s\dim_{\C}\M$ (\cref{CSRproperties}).
\end{proof}

\begin{prop}\label{Prop using gradings to prove QFi result}
Let $(\M,\Fi)$ be a weight-1 CSR. %
Then the $\Fi$-filtration satisfies
\mathcal{F}^{\Fi}_{\lambda<1}\subsetneq H^{\geq 2}(\M),
\qquad\quad
\mathcal{F}^{\Fi}_{1} = H^{\geq 2}(\M),
\qquad\quad \mathcal{F}^{\Fi}_{2}=H^*(\M),$$
and $H^{top}(\Sigma)[\dim_{\C}\M]$ kills the last surviving $H^{\dim_{\C}\M}(\M)$ class.
\end{prop}
\begin{proof}
By \cref{HitingTimeOfColumns}, only the $0$- and $1$-blocks, and the $1$- and $2$-pillars can hit $H^*(\M)$. So $\mathcal{F}^{\Fi}_{2}=H^*(\M)$.
The $1$-block and $2$-pillar lie in grading $\leq -1$, so
$H^{\geq 2}(\M)\subset \mathcal{F}^{\Fi}_{1}$.
It remains to show that $H^0(\M)$ was neither killed by the $0$-block nor by the $1$-pillar. For the $0$-block, this is immediate since it lies in grading $\geq 0$.
The $1$-pillar cannot kill the unit by \cref{Mid Coh of Hypersurface Vanish}, as
$H^*(\Sigma)[\dim_\C \M]$ in degree $-1$
$H^{\dim_\C \M-1}(\Sigma),$ which vanishes
by \cref{Mid Coh of Hypersurface Vanish}.
For the final claim, the top class of the $1$-pillar is in grading $\dim_\C \M-1$, and it must kill a class to the left by \cref{PropTwoRowSplitSpectralSeq}.
By \cref{Lemma block description spectral sequence}, the $0$-block lies in grading $\leq \dim_\C \M-1$. So it must kill a class in the $0$-pillar in $H^{\dim_{\C}\M}(\M)$.
\end{proof}
\section{Examples of spectral sequences}\label{ExamplesSpSeq}
We will illustrate the {\MBF } spectral sequences $\Epqr$ from \cref{SpecSeqMBF} for three classes of CSRs, and for Higgs moduli spaces.
Via \cref{FiltrationOnCohomologyBySpectralSequence}, we deduce the (rank-wise) filtrations of their ordinary cohomologies. %
\begin{rmk}{\bf Conventions.} We assume that all cohomologies are computed over the Novikov field $\k$, over a base field $\mathbb{B}$ of \textbf{characteristic zero} (\cref{Rmk about coeffs Novikov}), so \cref{CohomologyOfACSRProperties}, \cref{CalculationOfTheSliceCohomology} and \cref{Mid Coh of Hypersurface Vanish} apply. Interesting torsion phenomena arise in non-zero characteristics, but it is outside the scope of the current paper.
We continue with the conventions from  \cref{NotationForSpSeq} and \cref{Remark Conventions about the spectral sequence Figures}, but in the Figures we relabel $B_{p}$ to $B_{T_p}$, as $T_p$ helps indicate the torsion subgroup of the {\MB } submanifold.
In spectral sequence pictures, we include just the columns that may hit non-zero classes in the $0$-column via iterated differentials, as these determine the filtrations $\Fil^{\varphi}_\lambda.$ Sometimes we include additional columns beyond that. For CSRs we will frequently use the $\omega_{\C}$-duality fact from \eqref{NonDegenPairingByOmC}.
We abbreviate a field summand $\k$ in degree $m$ by
\k_m:= \k[-m].
\end{rmk}

\subsection{Springer resolutions }\label{ExamplesSpSeqSprRes}

Given a decomposition $p=(p_1,\dots,p_n)$ of $n,$\footnote{Meaning integers $p_i \geq 0$ with $p_1+\dots + p_n =n.$}
the associated $p$-partial flag variety is
$$\BP=\{F:=(0=F_0 \subset F_1 \subset \dots \subset F_{n-1} \subset F_n=\C^n) \mid \dim F_i/F_{i-1}=p_i\}.$$
If a decomposition $p$ is also a partition,\footnote{meaning all $p_i>0$ and they are weakly-descending ordered ($p_i\geq p_{i+1}$).} one also defines the adjoint orbit $\O_p \subset \sl_n$ of nilpotent matrices whose Jordan normal form decomposes into blocks according to $p.$\footnote{A nilpotent matrix has zero eigenvalues, so $p$ completely determines a Jordan normal form with block sizes $p_i$.}
Given a decomposition $p$, its associated \textbf{generalised Springer resolution} is the map
\begin{equation}\label{partialSpringer}
\nu_p: \NNN_p \fun \ol{\mathcal{O}_{p_+^*}},\qquad \{(e,F) \mid  F\in \B_p, \ e F_{i} \subset F_{i-1} (\forall i=1\dots n) \}\mapsto e.
\end{equation}
Here, $\mathcal{O}_{p_+^*}\subset \sl_n$ is the nilpotent adjoint orbit whose matrices have Jordan partition $p_+^*,$ where $p_+$ is a weakly-descending permutation of $p$ with $p_i=0$ removed and $p_+^*$ is its dual partition.\footnote{defined e.g. by flipping the corresponding Young diagrams.}

It is known that $\NNN_p \iso T^*\B_p$ (with projection to $\B_p$ given by $(e,F)\mapsto F$). %
Moreover, the map \eqref{partialSpringer} is known to be a \textbf{weight-1 CSR}, with the $\C^*$-action $\Fi$ that contracts the fibres on $T^* \B_p$ and
dilates matrices in $\ol{\mathcal{O}_{p_+^*}}.$
Thus, the action on $T^* \B_p$ is free outside the fixed locus $\B_p,$ %
hence there are no torsion points, so the spectral sequence $\Epqr$ is very simple.
The $0$-th column is the cohomology $H^*(\B_p)$ without a shift, whereas the $1$st and $2$nd columns are the cohomologies of the sphere bundle
$H^*(S(T^* \B_p))$ with shifts of $d:=\dim_{\R} \B_p$ and $2d$ rows downward, by \eqref{Equation shift down for integer time hypersurfaces} (using \cref{CohomologicalIndependenceOnHypersurface} and \cref{SpSeqIs1periodic}).
Recall $H^*(S(T^* \B_p))$ can be computed by \cref{CalculationOfTheSliceCohomology} and \cref{Mid Coh of Hypersurface Vanish}.

\begin{ex}\label{BabyBabyExample} $T^*\C P^1\fun \overline{\mathcal{O}_{11}}.$ (See \cref{SS_S11})

The first three columns\footnote{We number the columns starting from $0$, so we mean ``the first three columns after the 0-th column.''} are $H^*(S(T^* \CP^1))=H^*(\RP^3)=\k_0\oplus\k_{3}$ (recall $\mathrm{char}(\mathbb{B})=0$), %
with a shift down by $2$, $4$ and $6.$
The filtration $\Fil^{\varphi}_\lambda $ is ``rank-wise'' equal to:
$$0\subset \k_{2} \subset \k_0\oplus \k_{2} = H^*(T^*\C P^1).$$
So, writing $H^*(T^*\C P^1)=\k[a]/a^2,\ |a|=2,$ the filtration is $0 \subset \la a \ra \subset H^*(T^*\C P^1).$
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{S_11-cropped.pdf}
\caption{Spectral sequence for $T^*\C P^1$}
\label{SS_S11}
\end{figure}
\end{ex}

\begin{ex}\label{Example Tstar of flag variety}$T^* \B_p \fun \ol{\mathcal{O}_{p_+^*}}.$

The first three columns are $H^*(S(T^* \B_p))$ shifted down by $d$, $2d$ and $3d$ (where $d=\dim_{\R} \B_p$). Only the first two columns can hit the 0-th column via iterated differentials. The second column can hit only $H^0(T^* \B_p )$. As $H^*(S(T^* \B_p))$ has no middle degree classes, rows $0$ and $-1$ of the first column are zero, so no class in the first column can hit $1\in H^0(T^* \B_p ),$ thus it gets killed by the second column.
Altogether, the filtration is
\begin{equation}\label{filtr_flag_var}
0 \subset H^{\geq 2}(T^* \B_p) \subset H^*(T^* \B_p).
\end{equation}
For example, for the decomposition $p=(1,n)$ of $n+1$, we have $\B_p=\C\mathbb{P}^n$, $p_+^*=(2,1,1,\ldots,1)$ (with $n-1$ copies of $1$), and \eqref{partialSpringer} becomes $ T^*\C\mathbb{P}^n \fun \ol{\mathcal{O}_{21\dots 1}}=\{A\in \mathfrak{sl}_{n+1} \mid A^2=0, \ \rk(A)=1 \}$,
and the filtration is $0\subset H^{\geq 2}(\C\mathbb{P}^n) \subset H^*(\C\mathbb{P}^n)\cong H^*(T^*\C\mathbb{P}^n)$.
\end{ex}

\begin{rmk}\label{Remark all Springer resolutions}
The the same filtration \eqref{filtr_flag_var} holds for Springer resolutions of \textit{any} type, so $T^*(G/P),$ for $G$ reductive and $P$ parabolic, due to the verbatim argument, but
we expose type A here due to its use
%of its Springer resolution
for Slodowy varieties of type A, \cref{Examples Slodowy varieties}.
\end{rmk}

In order to get filtrations finer than \eqref{filtr_flag_var} on the cohomology of a flag variety,
we can compose the action $\Fi$ that contracts the fibres of $T^*\B_p$ with a 1-parameter subgroup $G=(G_t)_{t\in\C^*}$ of the maximal torus $T\leq GL(n,\C),$
as it induces a holomorphic $\C^*$-action on $T^*\B_p$ that commutes with $\Fi.$
The subgroups $G\leq T$ form a lattice isomorphic to $\Z^n,$ via the map
$$h=(h_1,\dots,h_n) \in \Z^n \mapsto t^h:=\diag(t^{h_1},\dots,t^{h_n}) \leq T.$$
However, to keep the composition of $\Fi$ and $G$ contracting, it reduces the choice to a finite convex subset of this lattice.
More precisely, fixing an integer $s>0$ there is a finite convex subset
$$K_s:=\{h \mid t^{h}\Fi^s \text{ is a contracting action on } T^*\B_p\} \subset \Z^n.$$
It is not hard to prove that
$K_1=\emptyset$, $K_s \subset K_{s'}$ for $s<s'$ and $\cup_s K_s = \Z^n.$
Moreover, we obtain:
\begin{prop} Given any integer $s>0,$ there is a map
$$K_s \fun \{\text{filtrations of } H^*(\B_p) \text{ by cup-ideals} \}, \ h \mapsto \mathcal{F}_\lambda^{t^h \Fi^s}.$$
\end{prop}

\begin{ex}\label{Example Tstar of flag variety with a twisted action}
\textbf{$T^* \CP^{n-1} \fun \ol{\mathcal{O}_{21 \dots 1}}$, with a twisted action.}

Let %
The action $t^h \Fi^n|_{\CP^{n-1}} =t^h \dejstvo \CP^{n-1}$ is $[z_0,t z_1, \dots , t^{n-1} z_{n-1}].$ %
The fixed points are $\F_i:=[0,\dots,1,\dots,0],\ i=0, \dots, n-1,$ where $1$ is in the $i$-th position.
There is a $(t\fun \infty)$ \CC-flowline from $\F_i$ to $\F_j$ if and only if $i<j,$ and it is $\Z/(j-i)$-torsion.
This way we get the weight decompositions of $\F_i$ in $\B_p.$ %
For example, the weights of $\F_{n-1}$ are $(-1,-2,\dots,-(n-1)).$
To get the full decompositions in $T^*\CP^{n-1}$, we use the $\om_\C$-pairing
$H_k \leftrightarrow H_{n-k}$ (\eqref{NonDegenPairingByOmC}, here the action has weight $s=n$)
This yields additional weights
$(n+1,n+2,\dots,2n-1)$ to $\F_{n-1}.$
\footnote{One can also compute the weights in coordinates. Near $\F_i\subset \C\P^{n-1}$ we have local coordinates $w_j=z_j/z_i$, for $j=0,1,\ldots,\widehat{i},\ldots,n-1$, whose weights are %
Those induce opposite weights on the cotangent fibre basis $dw_j$, but we also need to add weight $n$ coming from $\Fi^n$, so: $(n+i,n+i-1,\ldots,n+1,\widehat{n},n-1,\ldots,i+1)$. The latter are ``outer'' weights of $\F_i$: they yield torsion submanifolds that leave the core. For example, for $n=4$, writing ((core-weights),(outer-weights)), we have for $\F_0$: $((1,2,3),(3,2,1))$, $\F_1$: $((-1,1,2),(5,3,2))$, $\F_2$: $((-2,-1,1),(6,5,3))$, $\F_3$: $((-3,-2,-1),(7,6,5))$. For example, $\F_1$ is contained in a rank one $3$-torsion bundle, $\F_0$ is contained in a rank two $3$-torsion bundle,
and $\F_2$ is contained in a rank two $3$-torsion bundle caused by the weights $3,6$ (using that $\F_0,\F_1,\F_2$ are all $3$-minimal). This causes the $B_{1/3}$ column to contain the cohomology of a copy of $S^1$ and two copies of $S^3$.
we deduce the weights for the other $\F_i,$ and we observe that %
for each integer $k\in[n+1,2n-1]$ and $i\in [k-n,n-1]$, there is a 1-dimensional $k$-weight space in $\F_{i}$.
Each one of them yields a $\Z/k$-torsion line bundle $\mathcal{H}_k \fun \F_i$
in $T^*\CP^{n-1},$ whose corresponding {\MB } submanifold is $B_{1/k,i}\iso S^1.$
By \eqref{CZindecesOfMorseBottSumbanifolds},
Thus, for $c=1,\dots,n-1,$ the $c$-th column or equivalently, time$-\frac{1}{2n-c}$ column of the spectral sequence consists
of $c$ copies of $H^*(S^1)[2n-2-2c],$
and a certain linear combination of its $H^1(S^1)$-classes is killing the $H^{2n-2c}(\CP^{n-1})$ class in the $0$-th column.
There are no other columns between these, as $\frac{1}{n+1}<\frac{2}{2n-1}.$
Thus, these columns kill all the cohomology except for the unit $1\in H^0(\CP^{n-1}),$ which gets killed strictly after
time $\frac{1}{n+1}.$
Altogether, the filtration on $H^*(T^*\C P^{n-1})=\k[a]/a^n,\ |a|=2$ is
$$0 \subset \la a^{n-1} \ra \subset \la a^{n-2} \ra \subset \cdots \subset \la a \ra \subset H^*(T^*\C P^{n-1}),$$
so the Schubert cells in
$\CP^n$ are
killed by certain (linear combinations of) torsion $S^1$-orbits.
\cref{SS_TCP3tw} is the case $n=4.$
The \textbf{4-star symbol} is the point of local central symmetry (\cref{CentralSymmetrySpecSeq}). \textbf{Asterisk signs $*$ in the shifts} of some columns indicate that shifts of their connected components \textbf{are different.}
\begin{figure}[H]
\centering
\includegraphics[scale=0.7]{TCP3-cropped.pdf}
\caption{Spectral sequence for $T^*\C P^3$ with twisted action}
\label{SS_TCP3tw}
\end{figure}
\end{ex}

\subsection{Resolutions of Du Val singularities}\label{DuValSingularitiesSpSeq}

The CSRs of the lowest dimension are resolutions of ADE singularities\footnote{also known as simple surface singularities, Du Val singularities, Kleinian singularities or rational double points.} They are the minimal resolutions of quotient singularities
$$\pi_{\Gamma}:X_\Gamma\fun \C^2/\Gamma,$$
for any finite subgroup $\Gamma\leq SL(2,\C).$ Such groups $\Gamma$ (up to conjugation) are in bijection with ADE Dynkin graphs $Q_\Gamma$. This \textit{McKay graph} $Q_{\Gamma}$\footnote{more precisely, the quiver obtained by doubling the edges of $Q_\Gamma,$ and directing them in both ways.} encodes representation-theoretic properties of $\Gamma$.
The core $\pi_{\Gamma}^{-1}(0)$ is a union of 2-spheres intersecting transversely according to
$Q_\Gamma$.\footnote{The vertices of $Q_\Gamma$ correspond to the spheres; an edge in $Q_\Gamma$ corresponds to the intersection between the spheres.} Any $\C^*$-action on $X_\Gamma$ preserving the core will induce a %
$\C^*$-action on each sphere, so the intersection points of those spheres must lie in the fixed locus $\F$.
The Dynkin graph $A_n$ corresponds to the cyclic group $\Z/(n+1)$; the graph $D_n$ corresponds to the binary dihedral group $BD_{4(n-2)}$; the graphs $E_6,E_7$ and $E_8$ correspond to the binary tetrahedral, octahedral and icosahedral groups.

Klein showed that the coordinate ring $\C[\C^2/\Gamma]=\C[\C^2]^{\Gamma}$ is generated by three polynomials, hence the variety $\C^2/\Gamma$
embeds into $\C^3$ via a polynomial map $(z_1,z_2)\mapsto (f,g,h)\subset \C^3$ (see e.g. \cite[Sec.1.2]{Dol07}). Using $(X,Y,Z)$ as coordinates on $\C^3$, the embeddings for types A and D are:
\begin{equation}\label{ADEsingularities}
\begin{aligned}
& \Gamma=\Z/n: \ \ \ (z_1,z_2)\mapsto (z_1^{n},z_2^{n},z_1 z_2),\text{the image is } XY - Z^{n}=0,\\
& \Gamma=BD_{4n}:\ (z_1,z_2)\mapsto ((z_1^{2n}-z_2^{2n})z_1 z_2, z_1^{2n}+z_2^{2n} ,z_1^2 z_2^2), %
\text{the image is } X^2 - Y^2 Z + 4 Z^{n+1}=0.\\
\end{aligned}
\end{equation}

The natural {\contracting } weight-2 action on these spaces comes from the dilation action
\begin{equation}\label{ActionOnC2}
\C^*\dejstvo \C^2, \ t\cdot (z_1,z_2)=(t z_1,t z_2).
\end{equation}
Call \textbf{standard action} the induced action on $X_\Gamma\fun \C^2/\Gamma$.
\cref{BabyBabyExample} is the $A_1$-singularity.\\

We first consider type $A$. %
In the $\C^3$ coordinates of (\ref{ADEsingularities}), the standard action is
$$t\cdot X=t^{k} X,\ \ \text{  }t\cdot Y=t^{k} Y,\ \ \text{  }t\cdot Z=t^2Z.$$
The torsion subvarieties are the (complex) lines $l_1=(0,Y,0)$ and $l_2=(X,0,0),$
which have $\Z/k$-isotropy.
The torsion submanifolds in $X_{\Z/k}$ are the two $\Z/k$-torsion lines, $\widetilde{l_1}$ and
$\widetilde{l_2}$ that converge
to the two outer fixed points of the core.
For $p=-1,\dots,-(k-1)$, the intersections $B_{p,i}=\widetilde{l}_i\cap \Sp$ with each slice $\Sp$
yields two {\MB } submanifolds%
$$B_{p,1}\iso B_{p,2}\iso S^1.$$
By Lemma \ref{CohomologicalIndependenceOnHypersurface}, the hypersurface $B_k$ has $H^*(B_k)\cong H^*(S^3/(\Z/k))\cong H^*(S^3)\cong \k_0\oplus\k_{3}.$
As the core of $X_{\Gamma}$ is an $A_{k-1}$-tree of 2-spheres,
$H^*(X_{\Gamma})=\k_{0}\oplus \k_2^{k-1}.$
To compute the shifts $\mu(B_{p,i})$ we consider the weight decomposition at the
two outer fixed points.
The weight decomposition at any fixed point is $\C_k \oplus \C_{2-k},$ for some $k$, due to the duality $H_k \xleftrightarrow{\om_\C} H_{2-k}$ induced by the weight-2 action. The next step depends on the parity of $k$.

\begin{ex}\label{DuVal_odd}\textbf{$\Gamma=\Z/k,$ for odd  $k\geq 3$. ($A_{k-1}$-singularity)} %

For this parity, $\tfrac{k-1}{2}$ is an integer, and by (\ref{CZindecesOfMorseBottSumbanifolds}) the shifts are:
\mu(B_{p,1})=\mu(B_{p,2})=
1-\W(\tfrac{-p}{k} k)-\W(\tfrac{-p}{k}(2-k))
\begin{cases*}
-2, & $p<-\frac{k-1}{2},$\\
0, & $p\geq -\frac{k-1}{2}.$
\end{cases*}
Hence, the first $\frac{k-1}{2}$ columns of $E_1$ consist of two copies of $H^*(S^1)$; the next $\frac{k-1}{2}$ columns consist of two copies of $H^*(S^1)$ shifted down by 2; the next column is $H^*(S^3/(\Z/k))\cong H^*(S^3)$ shifted down by $2\mu=s\cdot \dim_{\C} X_{\Gamma}=2 \cdot 2 = 4$. All classes in the 0-th column $H^*(X_{\Gamma})$ will die, so they vanish two by two in each page, and $1\in H^0(X_{\Gamma})$ vanishes last on some page $E_r,$ where $r>k+\frac{1}{2}.$ By \cref{FiltrationOnCohomologyBySpectralSequence}  the filtration $\Fil^{\varphi}_\lambda$ is rank-wise
$$0\ \subset \ \k_2^2 \ \subset \ \k_2^4 \ \subset \cdots\ \subset \ \k_2^{k-1}\ \subset\  \k_{0}\oplus \k_2^{k-1}=H^*(X_{\Z/k}).$$
Figure \ref{SS_S41} shows the case $k=5$ (so $\frac{k-1}{2}=2$): we match by blue boxes the sets of classes that are killed one from the other by the green edge-differentials (in general we do not know whether these maps are diagonalisable).
The extended attraction graph for $k=5$ was discussed in [41] (the vertices correspond to five fixed points, and the outer green arrows are the directions of $\widetilde{l_1}$ and $\widetilde{l_2}$).

\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{S41.pdf}
\caption{Spectral sequence for $X_{\Z/5}$}
\label{SS_S41}
\end{figure}

\end{ex}

\begin{ex}\textbf{$\Gamma=\Z/k,$ for even $k\geq 2$.  ($A_{k-1}$-singularity)} %

This time the action is even. By considering the square root of the action, we see that the time-$\frac{1}{2}$ column of the $E_1$-page is  $H^*(\Sp)[2],$ and later columns repeat with shift $2\cdot 1=2$ (the square root of the action has weight $1$). The classes in $H^2(X_{\Gamma})$ vanish two by two until the last one is hit by the top class of the time-$\frac{1}{2}$ column, and $1\in H^0(X_{\Gamma})$ is killed on a later page.
The filtration $\Fil^{\varphi}_\lambda$ is
$$0\ \subset \ \k_2^2 \ \subset  \ \k_2^4 \ \subset  \cdots \ \subset \ \k_2^{k-2} \ \subset \ \k_2^{k-1}\  \subset \ \k_{0}\oplus \k_2^{k-1}=H^*(X_{\Z/k}).$$ \cref{SS_S51} shows the spectral sequence for $k=6.$

\begin{figure}[H]%
\centering
\includegraphics[scale=0.6]{S51.pdf}
\caption{Spectral sequence for $X_{\Z/6}$}
\label{SS_S51}
\end{figure}
\end{ex}

\begin{ex}\label{ExampleZ3Exotic}\textbf{$\Gamma=\Z/3,$ with a non-standard action.} %

We consider the non-standard weight-$2$ action induced by
$$t\cdot X=t^{2} X,\text{  }t\cdot Y=t^{4} Y,\text{  }t\cdot Z=t^2Z.$$
All points are $\Z/2$-torsion, and there is a $\Z/4$-torsion line $l_1=(0,Y,0).$ Its resolution $\widetilde{l_1}$ is a
$\Z/4$-torsion line that converges to a fixed point with weight decomposition $\C_4\oplus \C_{-2}.$ By (\ref{CZindecesOfMorseBottSumbanifolds}) the shift of the {\MB } submanifold $B_1=\widetilde{l_1} \cap \Sigma_1$ is $\mu(B_1)=0.$ As all points are $\Z/2$-torsion, $B_2=\Sigma_2$ and $\mu(B_2)=-2.$ By the symmetry in \cref{CentralSymmetrySpecSeq} we obtain the 3rd and 4th columns. For degree reasons, the other columns cannot hit $H^*(X_{\Gamma}).$ See Figure \ref{SS_S21Exotic}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{S21Exotic.pdf}
\caption{Spectral sequence for $X_{\Z/3}$ with a non-standard action.}
\label{SS_S21Exotic}
\end{figure}
The filtration $F^{\varphi}_\lambda$ is rank-wise
$$0\ \subset \ \k_{2} \ \subset \ \k_2^2 \ \subset \ \k_0\oplus \k_2^2 = H^*(X_{\Z/3}),$$
which is different from the filtration  obtained by the standard action in Example \ref{DuVal_odd}:
$$0\ \subset \ \k_2^2 \ \subset \ \k_0\oplus \k_2^2 = H^*(X_{\Z/3}).$$
So this example and Example \ref{DuVal_odd} for $k=3$ yield two different Floer-theoretic presentations of the ordinary cohomology %
of $X_{\Z/3},$ and determine two different filtrations by ideals on $H^*(X_{\Z/3})$.
\end{ex}

Now consider type D singularities. The action (\ref{ActionOnC2}) on a $D_{n+2}$-singularity, in the $\C^3$ coordinates of (\ref{ADEsingularities}), is
$t\cdot X=t^{2n+2} X$, $t\cdot Y=t^{2n} Y$, $t\cdot Z=t^4 Z.$
All points are $\Z/2$-torsion, so it is more convenient to work with the weight-1 action obtained by taking the square-root of the standard action:
\begin{equation}\label{SquareRootStandard}
t\cdot X=t^{n+1} X,\ \text{  }t\cdot Y=t^{n} Y,\ \text{  }t\cdot Z=t^2 Z.
\end{equation}

\begin{ex} \textbf{$\Gamma=BD_{8k}$ ($D_{2k+2}$-singularity), $k\geq 1.$}\label{ExampleD_nSingEven}
(See \cref{D6_sp_seq})\vspace{1mm}

For $n=2k$, (\ref{SquareRootStandard}) is
$t\cdot X=t^{2k+1} X$, $t\cdot Y=t^{2k} Y$, $t\cdot Z=t^2 Z.$
There are three torsion lines
$$\a=(0,Y,0),\ \b_1=(0,2Z^k,Z),\ \b_2=(0,-2Z^k,Z),$$
with respective torsion groups $\Z/2k,\Z/2,\Z/2$.
As it is a weight-1 action, it has a minimal component, which must be the sphere that corresponds to the unique trivalent vertex of the graph $D_{2k+2}.$\footnote{Because that sphere carries three intersection points fixed by the action, so the $\C^*$-action must fix the whole sphere.}
The resolutions $\tilde{\a},\tilde{\b_1},\tilde{\b_2}$ of the torsion lines $\a,\b_1,\b_2$ are lines again, and they converge to the fixed points of the spheres that correspond to the leaves of the graph $D_{2k+2}$ (because these are the leaves of the attraction graph, and by Proposition \ref{THERE_ARE_ISOTROPIES} each such leaf has a torsion bundle that converges to it).

Using formula (\ref{CZindecesOfMorseBottSumbanifolds}), as in the previous examples, one can calculate the degree shifts $\mu(B_{p,c})$ of the Morse Bott submanifolds $B_{p,c}\cong S^1$ which are intersections of the lines $\tilde{\a},\tilde{\b_1},\tilde{\b_2}$ with the slices $\Sp$. They all have zero shift, for $T_p<1.$ Then, $B_1=\Sigma_1$ has a shift down by 2 (as the action has weight 1), and the other columns are determined by  Corollary \ref{SpSeqIs1periodic}.
The filtration $F^{\varphi}_\lambda $ is rank-wise:
\begin{align*}
&\text{for } k=1: \ 0\subset \k_2^3 \subset \k_2^4 \subset \k_{0}\oplus \k_2^{4}=H^*(X_{\Gamma}),\\
&\text{for } k\geq 2: \ 0\subset \k_{2} \subset \dots \subset \k_2^{k-1} \subset \k_2^{k+2} \subset \k_2^{k+3}\\  &\hspace{4.1cm}  \dots \subset \k_2^{2k+1} \subset \k_2^{2k+2} \subset H^*(X_{\Gamma}).
\end{align*}
In both cases the jump in rank by $3$ occurs at the time-$\frac{1}{2}$ column, as all three lines are $\Z/2$-isotropic. %
\begin{figure}[H]
\centering
\includegraphics[scale=0.60]{D6-cropped.pdf}
\caption{Spectral sequence for $D_6$ ($k=2$, $n=4$).}
\label{D6_sp_seq}
\end{figure}
\end{ex}

\begin{ex} \textbf{$\Gamma=BD_{8k+4}$ ($D_{2k+3}$-singularity), $k\geq 1.$}\label{ExampleD_nSingOdd} (See \cref{D5_sp_seq})\vspace{1mm}

For $n=2k+1$, (\ref{SquareRootStandard}) is
$t\cdot X=t^{2k+2} X$, $t\cdot Y=t^{2k+1} Y$, $t\cdot Z=t^2 Z.$
There are three torsion lines
$$\a=(0,Y,0),\ \b_1=(2i Z^{k+1},0,Z),\ \b_2=(-2i Z^{k+1},0,Z),$$
with respective isotropies $\Z/(2k+1),\Z/2,\Z/2$. Again they give rise to circles $B_{p,\c}$ with zero shifts. The novelty now is the time $\frac{1}{2}$-column: there are \emph{two} {\MB } submanifolds, so the filtration $F^{\varphi}_\lambda $ at time $\frac{1}{2}$ jumps by rank 2 in degree 2. So the filtration is rank-wise
\begin{equation*}
0\subset \k_{2} \subset \dots \subset \k_2^{k} \subset \k_2^{k+2} \subset \k_2^{k+3} \subset \dots \subset \k_2^{2k+3} \subset H^*(X_{\Gamma}).
\end{equation*}

\begin{figure}[H] %
\centering
\includegraphics[scale=0.6]{D5.pdf}
\caption{Spectral sequence for $D_5$ ($k=1$, $n=3$).}
\label{D5_sp_seq}
\end{figure}
\end{ex}

\begin{ex}\textbf{$D_n$ via the attraction graph.} \label{Dn via extended attraction graph}
It soon becomes desirable to obtain descriptions of the {\MB } manifolds that avoid coordinate presentations like (\ref{ADEsingularities}), instead using the attraction graph (\cref{Subsection Attraction graphs}).
We illustrate this for $D_n$.
The core consists of a Dynkin tree $D_n$, each vertex corresponds to a sphere, each edge corresponds to an intersection of spheres which in turn is a fixed point of the action,
and the sphere $C$ at the unique trivalent vertex is fixed (as it carries three fixed points).
At each fixed point, $\omega_{\C}$-duality implies that the weight decomposition is $H_k\oplus H_{2-k}$, each summand having $\dim_{\C}=1$.
If a sphere in the core is fixed, it has a weight $k=0$, thus $H_0\oplus H_2$ and the {\MB } index is $0$. By \cref{EqnFrankel} there can be only one fixed sphere, which is $C$, so the other spheres must carry exactly two fixed points (by considering the rank in \cref{EqnFrankel}).
By \cref{CorollaryGradientTrajBecomeSpheres}, it is now easy to write out the weight decompositions at all fixed points, and the attraction graph is again a $D_n$-Dynkin graph.
It is preferable to use the square-root of the action, %
which exists as there is a fixed component $C$ of maximal ($\om_\C$-Lagrangian) dimension in the core.
This is based on the analytic continuation of the square-root that exists on the torsion bundle
$\mathcal{H}_2\fun C,$
see \cite[Lem.5.4. and Prop.5.5]{vzivanovic2022exact}.
Now, as the square-root is a weight-1 {\contracting } action,
$\omega_{\C}$-duality gives $H_k\oplus H_{1-k}$ decompositions.
We have $H_0\oplus H_1$ at $C$, e.g.\;at intersections points $p_1,p_2,p_3$ with the adjacent spheres $C_1,C_2,C_3$; at the second fixed points $q_1,q_2,q_3$ of $C_1,C_2,C_3$ we get $H_{-1}\oplus H_2$ by \cref{CorollaryGradientTrajBecomeSpheres} ($C_i$ defines a $\C^*$-flow from $p_i$ to $q_i$).
For $D_4$, the $q_i$ are leaf nodes in the attraction graph, and their $H_2$ defines a $\Z/2$-torsion line bundle $\mathcal{H}_2$, yielding {\MB } manifolds $S^1$ when intersecting with slices.
For $D_n$ with $n>4$, the same applies to two of those $q_i$. The third corresponds to where additional edges arise in the $D_n$ Dynkin diagram: they correspond to fixed points with $H_{-2}\oplus H_3$, $H_{-3}\oplus H_4$, $\ldots$, $H_{-(n-3)}\oplus H_{n-2},$ yielding a $\Z/(n-2)$-torsion line bundle at the leaf (and so again $S^1$ {\MB } manifolds). So we recover Examples \ref{ExampleD_nSingEven} and \ref{ExampleD_nSingOdd} without needing (\ref{ADEsingularities}).
\end{ex}
\begin{ex}\textbf{$E_6$, $E_7$, $E_8$ Singularities.} \label{Example E_678}(See Figure %
\ref{E6E7E8})

We use the method from \cref{Dn via extended attraction graph}, as the previous method that uses the invariant polynomials becomes too complicated.
The core for $E_6$ is the $E_6$-Dynkin tree of spheres, and the sphere $C$ that corresponds to the trivalent node of $E_6$ is fixed.
Thus, like in the previous example, %