\section{Enhancement to the bath due to intermediately coupled spins}

\label{app:eta}

In this section, we recap and extend the calculation of Ref.~\cite{crowley2022partial}. We derive the entropic enhancement to the ergodic grain due to multiple intermediately coupled spins. For simplicity we assume $H_\mathrm{g}$ to be a GOE matrix, however the results are readily generalizable to the case of an ETH satisfying system (see Ref.~\cite{crowley2022partial}). The results are obtained using an ansatz for the many body eigenstates which accounts for the effect of resonances. The main results of this appendix are:
\begin{enumerate}
    \item For an $N=2$ spin central grain model, where $g_{1,0}$ and $g_{2,0}$ are in the intermediate and weak coupling regimes respectively, the reduced coupling of the second spin is given by 
    \begin{equation}
        g_{2} = g_{2,0} \exp\left(\tfrac12 \eta(g_1)\right)
    \end{equation}
    where $g_1 = g_{1,0}$, and the entropic enhancement $\eta(g)$ is given by~\eqref{eq:eta_app}.
    \item For the general case of $N$ intermediately coupled spins, and a single $(N+1)$th spin in the weak coupling regime, the reduced coupling of the weakly coupled spin is enhanced by an additive entropic term
    \begin{equation}
        g_{N+1} = g_{N+1,0} \exp\left(\tfrac12 \sum_{n=1}^N\eta(g_n)\right).
    \end{equation}
    where, as before, $\eta(g)$ is given by~\eqref{eq:eta_app}.
\end{enumerate}
The form of $\eta(g)$ derived in this appendix~\eqref{eq:eta_app} is used in numerical calculations throughout this manuscript, and plotted in Fig~\ref{Fig:1}. 

\subsection{Single spin}

We begin by recapitulating the properties of the many body eigenstates in the $N=1$ case, studied in detail in Ref.~\cite{crowley2022partial}. 

When spin is in the intermediate coupling regime, the infinite time properties (i.e. distribution of entanglement entropies and time averaged autocorrelators) of the spin are determined by the distribution of the quantity $g_{1a}$---the norm of the first corrections in perturbation theory as $V_1$ is introduced~\eqref{eq:g1a}. As in the main text, we denote the distribution of this quantity with $p_{g_{1a}}(g)$ so that the ensemble averaged fraction of the $g_{1a}$ in the interval $[g , g + \d g]$ is given by $p_{g_{1a}}(g) \d g$. For the GOE grain considered here $p_{g_{1a}}(g)$ may be calculated
\begin{equation}
    \begin{aligned}
        p_{g_{1a}}(g) &= \tilde{p}(g/g_1)/g_1
        \\
        \tilde{p}(x) & = \frac{2}{x^2} \exp \left(- \frac{\pi^3}{4x^2}\right) \bigg( 1 + \frac{c_1}{x} + \frac{c_2}{x^2}  +O\left(x^{-3}\right)\bigg)
    \end{aligned}
    \label{eq:p_goe}
\end{equation}
where $c_1 = 5.3\ldots$, $c_2 = 11.2\ldots$. However, the main features are more general: (i) a rapid decay for $g \lesssim g_1$ (ii) a unimodal peak around the typical value $g \approx g_1$ and (iii) a power law tail at large $g$ given asymptotically by
\begin{equation}
    p_{g_{1a}}(g) \sim 2g_1/g^2.
\end{equation}
Here $g_1$ is defined (as in~\eqref{eq:g1}) by
\begin{equation}
    g_1 = \rho [|V_{1,ab}|].
\end{equation}
where $\rho$ is the many body density of states at maximum entropy, and $V_{1,ab}$ are the matrix elements of $V_1$ between the eigenstates $\ket{E_a}$ of $H$ evaluated for $V_1 = 0$, and the square brackets $[\cdot]$ denotes averaging over $a \neq b$. Each of these eigenstates is a product state of the first spin and grain
\begin{equation}
    \ket{E_a} = \ket{\epsilon_\alpha}\ket{\sigma}
    \label{eq:app_eig}
\end{equation}
for $a = (\alpha,\sigma)$, $\sigma \in \{ \uparrow , \downarrow \}$, and $\ket{\epsilon_\alpha}$ an eigenstate of the grain $H_\mathrm{g}$.

Consider following the eigenstates as the coupling $V_1$ tuned from zero to its finite value. As the spin is in the intermediate coupling regime, typically eigenstates are only perturbatively altered. However, a minority of states $\ket{E_a}$ are accidentally close to states from the opposite spin sector, and have correspondingly large values $g_{1a} \gtrsim 1$. These states strongly hybridize, forming resonances (depicted in Fig.~\ref{Fig:1}b). Typically, such resonances involve only a pair of states: the state $\ket{E_a}$, and some other nearby state $\ket{E_c}$. We may thus approximate $g_{1a}$
\begin{equation}
    g_{1a} = \sqrt{\sum_{b \neq a} \left| \frac{V_{1,ab}}{E_a - E_b} \right|^2 } \approx \left| \frac{V_{1,ac}}{E_a - E_c} \right|
    \label{eq:g1a_approx}
\end{equation}
where $c$ is obtained by minimising the denominator $|E_a - E_b|$.

An ansatz for the eigenstates is obtained by diagonalizing within these two-level resonance sub-spaces. The effective two-level Hamiltonian is given by
\begin{equation}
    H_\mathrm{eff} = \begin{pmatrix}
    E_a & V_{1,ac}
    \\
    V_{1,ac} & E_c
    \end{pmatrix} 
    \approx
    \Delta_{ac} \begin{pmatrix}
    1 & g_{1a}
    \\
    g_{1a} & 0
    \end{pmatrix} + E_c
    \label{eq:Heff}
\end{equation}
where $\Delta_{ac} = E_a - E_c$ and we have used~\eqref{eq:g1a_approx} that $g_{1a} \approx | V_{1,ac}/\Delta_{ac} |$. Explicit diagonalization of~\eqref{eq:Heff} yields the new eigenvectors
\begin{equation}
    \begin{aligned}
    \ket{E_a'} & = \sqrt{Q_{a}} \ket{E_a} + \sqrt{P_{a}} \ket{E_c}
    \\
    \ket{E_c'} & = \sqrt{P_{a}} \ket{E_a} - \sqrt{Q_{a}} \ket{E_c}
    \end{aligned}
    \label{eq:eig_1_spin}
\end{equation}
where $P_a = P(g_{1a})$, $Q_a = Q(g_{1a})$ are given by
\begin{equation}
    P(g) := 1 - Q(g) := \frac12 \left( 1 - \frac{1}{\sqrt{1 + 4 g^2}} \right).
    \label{eq:pq_def}
\end{equation}

\subsubsection*{Infinite time observables}
\label{app:inf_time}

This eigenstate ansatz~\eqref{eq:eig_1_spin} can be used to calculate the distribution of eigenstate entanglement entropies and infinite time memory of the spin, and agrees with numerics~\cite{crowley2022partial}. 

Specifically, we have that the entanglement entropy $S_{na}$ of the $(n=1)$th spin the $a$th state is given by $S_{na} = S(g_{na})$ with
\begin{equation}
    \begin{aligned}
    S(g) & = - P(g) \log P(g) - Q(g) \log Q(g)
    \end{aligned}
\end{equation}
and thus follows a distribution
\begin{equation}
    p_{S_{na}}(S_n) = \int \d g \delta(S_n - S(g)) p_{g_{na}}(g)
\end{equation}
with mean value
\begin{equation}
    [S] = \int \d g S(g) p_{g_{na}}(g).
\end{equation}
By similar arguments the infinite time correlator is given by
\begin{equation}
    \overline{C_n^{zz}} = \int \d g p_{g_{na}}(g) (P(g)-Q(g))^2.
\end{equation}
In the intermediate regime it is sufficient to write
\begin{equation}
    p_{g_{na}}(g) = g_n / g^2 + O(g_n^2/g^3)
\end{equation}
to obtain the limits given in the main text~\eqref{eq:S_Czz} (details in Ref.~\cite{crowley2022partial}).

\subsection{Two spins}

We next consider introducing a second spin. Again, this case was considered in Ref.~\cite{crowley2022partial}, and we here recap the calculation. The `effective bath' seen by the second spin is composed of the first spin and grain. As a result, the reduced coupling is enhanced from its bare ($V_1=0$) value
\begin{equation}
    g_{2} = g_{2,0} \exp\left(\tfrac12 \eta(g_1)\right)
    \label{eq:g2_app}
\end{equation}
in this section we calculate $\eta(g)$.

We begin by considering $p_{g_{2a}}(g)$, the distribution of $g_{2a}$ the norm of the first order term in perturbation theory when the eigenstate $\ket{E_a}$ is expanded about $V_2 = 0$, but for $V_1$ finite. This distribution has the same qualitative features as $p_{g_{1a}}(g)$: (i) a rapid decay for $g \lesssim g_2$ (ii) a unimodal peak around the typical value $g \approx g_2$ and (iii) a power law tail at large $g$ given asymptotically by
\begin{equation}
    p_{g_{2a}}(g) \sim 2g_2/g^2
\end{equation}
where 
\begin{equation}
    g_2 = \rho [|V_{2,ab}'|].
    \label{eq:g2_app2}
\end{equation}
where $V_{2,ab}'$ are the matrix elements of $V_2$ between the eigenstates $\ket{E_a'}$ of $H$ evaluated for $V_2 = 0$ but $V_1$ finite. Each of these eigenstates is a product state of the second spin and grain, but in general is an entangled state of the first spin and grain~\eqref{eq:eig_1_spin}. From~\eqref{eq:g2_app} and~\eqref{eq:g2_app2} it follows that
\begin{equation}
    \eta(g_1) = 2 \log \left(\frac{{[|{V_{2,ab}^{\mathrm{\prime}}}|]}}{[|V_{2,ab}|] }\right)
    \label{eq:eta_app2}
\end{equation}
where $V_{2,ab}$ are the matrix elements of $V_2$ between the eigenstates $\ket{E_a}$ of $H$ evaluated for $V_2 = 0$ and $V_1 = 0$. In the remainder of this section, we evaluate~\eqref{eq:eta_app2}.

Each many body eigenstate may be identified with a sector of the first spin by following the state adiabatically as we tune $V_1 \to 0$, and measuring the state of the first spin. Using this labelling scheme, we see that there are two qualitatively different families of matrix elements: $V_{2,ab}'$ is \emph{even} if $\ket{E_a'}$ and $\ket{E_b'}$ correspond to the same sector, and \emph{odd} otherwise. Half of the matrix elements are even, and half odd, so that
\begin{equation}
    [|V_{2,ab}'|] = \frac12 \left( [|{V_{2,ab}^{\mathrm{(e)}\prime}}|] + [|{V_{2,ab}^{\mathrm{(o)}\prime}}|] \right).
    \label{eq:VOE}
\end{equation}
where $[|{V_{2,ab}^{\mathrm{(e)}\prime}}|]$, $[|{V_{2,ab}^{\mathrm{(o)}\prime}}|]$ are the averaged taken over the odd/even sectors only. We evaluate each of these contributions in turn.

We first evaluate the even sector. We consider two generic states
\begin{subequations}
    \begin{align}
    \ket{E_a'} & 
    = \sqrt{Q_{a}} \ket{E_a} + \sqrt{P_{a}} \ket{E_c} 
    \nonumber
    \\
    & = \sqrt{Q_{a}} \ket{\epsilon_\alpha}\ket{\uparrow_1} + \sqrt{P_{a}} \ket{\epsilon_\beta}\ket{\downarrow_1}
    \label{eq:1spin_eig}
    \\
    \ket{E_b'} & 
    = \sqrt{Q_{b}} \ket{E_b} + \sqrt{P_{b}} \ket{E_d}
    \nonumber
    \\
    & = \sqrt{Q_{b}} \ket{\epsilon_\gamma}\ket{\uparrow_1} + \sqrt{P_{b}} \ket{\epsilon_\delta}\ket{\downarrow_1}
    \end{align}
\end{subequations}
where in each case in the second line we have denoted the state of the first spin $\sigma_1$ explicitly, following~\eqref{eq:app_eig}. As $V_2$ does not act on the first spin (i.e. $[V_2,\sigma_1^\alpha]=0$), the corresponding matrix element contains only two terms
\begin{equation}
    \begin{aligned}
        {V_{2,ab}^{\mathrm{(e)}\prime}} & = \sqrt{Q_a Q_b} V_{2,ab}^{\mathrm{(e)}} + \sqrt{P_a P_b} V_{2,cd}^{\mathrm{(e)}} 
    \end{aligned}
\end{equation}
where $V_{2,ab}^{\mathrm{(e)}}$ are the even matrix elements of $V_2$ calculated in the basis $\ket{E_a}$, the eigenbasis calculated for both $V_2 = 0$ \emph{and} $V_1 = 0$. Note that by this definition the odd elements are zero $V_{2,ab}^{\mathrm{(o)}} = 0$.

As the $P_a = P(g_{1a})$ describe resonances induced by $V_1$, they are uncorrelated with the matrix elements $V_{2,ab}$. Moreover, as the bare grain is ergodic, the matrix elements $V_{2,ab}$, $V_{2,cd}$ are iid Gaussian distributed. For iid Gaussian distributed random variables $x,x'$ with zero mean $[x] = [x']=0$, it follows have that $[|a x + b x'|] = \sqrt{a^2 + b^2}[|x|]$. Using this, we obtain
\begin{equation}
    \begin{aligned}
    {[|{V_{2,ab}^{\mathrm{(e)}\prime}}|]} & = [|V_{2,ab}^{\mathrm{(e)}}|] [\sqrt{P_a P_b + Q_a Q_b}]
    \\
    & = 2[|V_{2,ab}|] [\sqrt{P_a P_b + Q_a Q_b}].
    \end{aligned}
\end{equation}
where in the second line we have used that $[|V_{2,ab}|] = \tfrac12 ( [|{V_{2,ab}^{\mathrm{(e)}}}|] + [|{V_{2,ab}^{\mathrm{(o)}}}|] ) = \tfrac12 [|{V_{2,ab}^{\mathrm{(e)}}}|] $. Furthermore, $P_a$ and $P_b$ are uncorrelated, so we obtain
\begin{equation}
    \begin{aligned}
    {[|{V_{2,ab}^{\mathrm{(e)}\prime}}|]} & = 2[|V_{2,ab}|] \iint \d g \d g' p_{g_{1a}}(g) p_{g_{1a}}(g') K^{\mathrm{(e)}} (g,g')
    \\
    K^{\mathrm{(e)}} (g,g') & =  \sqrt{P(g) P(g')+ Q(g) Q(g')} 
    \end{aligned}
    \label{eq:Ke}
\end{equation}
Repeating the same series of arguments for the odd terms we obtain
\begin{equation}
    \begin{aligned}
    {[|{V_{2,ab}^{\mathrm{(o)}\prime}}|]} & = 2[|V_{2,ab}|] \iint \d g \d g' p_{g_{1a}}(g) p_{g_{1a}}(g') K^{\mathrm{(o)}} (g,g')
    \\
    K^{\mathrm{(o)}} (g,g') & =  \sqrt{P(g) Q(g')+Q(g) P(g')} 
    \end{aligned}
    \label{eq:Ko}
\end{equation}
By substituting~\eqref{eq:Ke} and~\eqref{eq:Ko} into~\eqref{eq:VOE} we have
\begin{equation}
    \begin{aligned}
    {[|{V_{2,ab}^{\mathrm{\prime}}}|]} & = [|V_{2,ab}|] \iint \d g \d g' p_{g_{1a}}(g) p_{g_{1a}}(g') K (g,g')
    \\
    K (g,g') & = K^{\mathrm{(e)}} (g,g') + K^{\mathrm{(o)}} (g,g')
    \end{aligned}
    \label{eq:V2prime}
\end{equation}
Substituting~\eqref{eq:V2prime} into~\eqref{eq:eta_app} we obtain the entropic enhancement to the effective bath due to hybridization between the grain and first spin
\begin{equation}
    \eta(g) = 2 \log\left( \iint \d g \d g' p_{g_{1a}}(g) p_{g_{1a}}(g') K (g,g') \right).
    \label{eq:eta_app}
\end{equation}
where the right-hand side depends on $g_1$ via its appearance in $p_{g_{1a}}$~\eqref{eq:p_goe}.

\subsection{Three spins}

We next consider introducing a third spin. To calculate $g_3$ we require a model for the eigenstates of the central grain model for finite $V_1$ and finite $V_2$. We generalize the ansatz~\eqref{eq:eig_1_spin} to the two spin case
\begin{equation}
    \begin{aligned}
    \ket{E_a''} & = \sqrt{Q_{a} Q_{a}'} \ket{E_a} + \sqrt{P_{a} Q_{a}'} \ket{E_b}
    \\
    & \quad 
    + \sqrt{Q_{a} P_{a}'}\ket{E_c} +
    \sqrt{P_{a} P_{a}'} \ket{E_d}
    \end{aligned}
    \label{eq:2spin_eig}
\end{equation}
where $P_{a} = 1 - Q_{a} = P(g_{1a})$ and $P_{a}' = 1 - Q_{a}' = P(g_{2a})$ $\sigma_n \in \{ \uparrow, \downarrow \}$ and we use a bar notation to denote spin flips so that $\bar{\uparrow} = \downarrow$, $\bar{\downarrow} = \uparrow$, and the $\ket{E_a}, \cdots , \ket{E_d}$ are eigenstates in the limit $V_1,V_2\to 0$, i.e. product states of the two spins
\begin{equation}
    \begin{aligned}
    \ket{E_a} & = \ket{\epsilon_\alpha}\ket{\sigma_1}\ket{\sigma_2},
    & 
    \ket{E_b} & =
    \ket{\epsilon_\beta}\ket{\bar{\sigma}_1}\ket{\sigma_2} ,
    \\ 
    \ket{E_c} & = \ket{\epsilon_\gamma}\ket{\sigma_1}\ket{\bar{\sigma}_2},
    &
    \ket{E_d} & = \ket{\epsilon_\delta}\ket{\bar{\sigma}_1}\ket{\bar{\sigma}_2}.
    \end{aligned}
\end{equation}

The calculation of $[|V_{3,ab}''|]$ then proceeds in direct generalization of the previous section. However, now there are more `species' of matrix element. Consider adiabatically following the eigenstates as we take the limit of $g_1,g_2 \to 0$: the state~\eqref{eq:2spin_eig} tends to the  $(\sigma_1,\sigma_2)$ sector of the two spins. As before, we use a convention in which we associate the state~\eqref{eq:2spin_eig} with the $(\sigma_1,\sigma_2)$ sector even for finite $V_1$, $V_2$. Consequently, there are four species of matrix element $V_{3,ab}''$ depending on whether $\sigma_1$, $\sigma_2$, both or neither must changed to relate the states $\ket{E_a''}$ and $\ket{E_b''}$, which we refer to as the (odd,even), (even,odd), (odd,odd) and (even,even) sectors respectively. 

The mean matrix element is obtained by averaging across these sectors
\begin{equation}
    \begin{aligned}
    {[|V_{3,ab}''|]} = & \frac14 \left( [|{V_{3,ab}^{\mathrm{(e,e)}\prime\prime}}|] +
    [|{V_{3,ab}^{\mathrm{(e,o)}\prime\prime}}|]  \right.
    \\
    & \quad \left. +
    [|{V_{3,ab}^{\mathrm{(o,e)}\prime\prime}}|] +
    [|{V_{3,ab}^{\mathrm{(o,o)}\prime\prime}}|] \right).
    \label{eq:VOOEE}
    \end{aligned}
\end{equation}
These are calculated following the same prescription of the previous section. In the (even,even) case we consider the matrix element between two states, e.g. $\ket{E_a''}$ from~\eqref{eq:2spin_eig} and 
\begin{equation}
    \begin{aligned}
    \ket{E_e''} & = \sqrt{Q_{a} Q_{a}'} \ket{E_e} + \sqrt{P_{a} Q_{a}'} \ket{E_f}
    \\
    & \quad 
    + \sqrt{Q_{a} P_{a}'}\ket{E_g} +
    \sqrt{P_{a} P_{a}'} \ket{E_h}
    \end{aligned}
\end{equation}
where similarly
\begin{equation}
    \begin{aligned}
    \ket{E_e} & = \ket{\epsilon_\epsilon}\ket{\sigma_1}\ket{\sigma_2},
    & 
    \ket{E_f} & =
    \ket{\epsilon_\zeta}\ket{\bar{\sigma}_1}\ket{\sigma_2} ,
    \\ 
    \ket{E_g} & = \ket{\epsilon_\eta}\ket{\sigma_1}\ket{\bar{\sigma}_2},
    &
    \ket{E_h} & = \ket{\epsilon_\theta}\ket{\bar{\sigma}_1}\ket{\bar{\sigma}_2}.
    \end{aligned}
\end{equation}
Computing the matrix element directly, we use that $V_3$ does not alter the state of the first or second spin, and obtain
\begin{widetext}
\begin{equation}
     {V_{3,ae}^{\mathrm{(e,e)}\prime\prime}} = \sqrt{Q_a Q_a ' Q_{e} Q_{e}'} \bra{E_a } V_3 \ket{ E_e } +  
     \sqrt{P_a Q_a ' P_{e} Q_{e}'}  \bra{E_b } V_3 \ket{ E_f } +
     \sqrt{Q_a P_a ' Q_{e} P_{e}'} \bra{E_c } V_3 \ket{ E_g } +
     \sqrt{P_a P_a ' P_{e} P_{e}'} \bra{E_d } V_3 \ket{ E_h } 
\end{equation}
with a mean size
\begin{align}
    {[|V_{3,ae}^{\mathrm{(e,e)}\prime\prime}|]} & = [|V_{3,ae}^{\mathrm{(e,e)}}|] [\sqrt{ P_a P_a ' P_{e} P_{e}' + Q_a P_a ' Q_{e} P_{e}' + P_a Q_a ' P_{e} Q_{e}' + Q_a Q_a ' Q_{e} Q_{e}' } ]
    \nonumber
    \\
    & = [|V_{3,ae}^{\mathrm{(e,e)}}|] [\sqrt{ P_a P_a ' + Q_a Q_a '} \sqrt{  P_{e} P_{e}' +  Q_{e} Q_{e}' } ]
    \nonumber
    \\
    & = 4 [|V_{3,ae}|] [\sqrt{ P_a P_a ' + Q_a Q_a '}][ \sqrt{  P_{e} P_{e}' +  Q_{e} Q_{e}' } ]
    \nonumber
    \\
    & = 4 [|V_{3,ae}|] \left(\iint \d g \d g' \ p_{g_{1a}}(g) p_{g_{1a}}(g') K^{\mathrm{(e)}} (g,g') \right) \left(\iint \d g \d g' \ p_{g_{2a}}(g) p_{g_{2a}}(g') K^{\mathrm{(e)}} (g,g') \right)
\end{align}
Repeating this calculation for the other sectors we obtain
\begin{equation}
    \begin{aligned}
    {[|V_{3,ae}^{(s,s')\prime\prime}|]} &  = 4 [|V_{3,ae}|] \left(\iint \d g \d g' \ p_{g_{1a}}(g) p_{g_{1a}}(g') K^{(s)} (g,g') \right) \left(\iint \d g \d g' \ p_{g_{2a}}(g) p_{g_{2a}}(g') K^{(s')} (g,g') \right)
    \end{aligned}
\end{equation}
for $s,s'\in \{ \mathrm{o},\mathrm{e}\}$ and hence, using~\eqref{eq:VOOEE}.
\begin{equation}
    \begin{aligned}
    {[|V_{3,ae}^{\prime\prime}|]} &  =  [|V_{3,ae}|] \left(\iint \d g \d g' \ p_{g_{1a}}(g) p_{g_{1a}}(g') K (g,g') \right) \left(\iint \d g \d g' \ p_{g_{2a}}(g) p_{g_{2a}}(g') K (g,g') \right)
    \end{aligned}
\end{equation}
\end{widetext}
With this result, we determine that the enhancement is given by a sum of the enhancements due to the two spins.
\begin{equation}
    \log \left( \frac{g_3}{g_{3,0}} \right) = \log \left(\frac{{[|{V_{3,ab}^{\mathrm{\prime\prime}}}|]}}{[|V_{3,ab}|] }\right) = \tfrac12 \left( \eta(g_1) + \eta(g_2) \right).
\end{equation}

\subsection{$N$ spins}

The many spin case is found by direct generalization of the previous section. Here we simply state the eigenstate ansatz, and the result. 

The ansatz for an eigenstate associated to the sector $\vec{\sigma} = (\sigma_1,\sigma_2 \cdots \sigma_N)$ is given by
\begin{equation}
    \begin{aligned}
    \ket{E_a^{(n)}} & = \sum_{\vec{\tau}} C_{\vec{\sigma},\vec{\tau}} \ket{\epsilon_{\alpha,\vec{\tau}}}\ket{\tau_1}\ket{\tau_2}\cdots \ket{\tau_N} 
    \end{aligned}
    \label{eq:Nspin_eig}
\end{equation}
where 
\begin{subequations}
\begin{align}
    C_{\vec{\sigma},\vec{\tau}} & = \prod_{n} c_{\sigma_n,\tau_n} 
    \\
    c_{\sigma_n,\tau_n} & = \begin{cases}
    \sqrt{ P(g_{na}) } \quad & \text{if} \quad \sigma_n = \tau_n
    \\
    \sqrt{ Q(g_{na}) } \quad & \text{if} \quad \sigma_n \neq \tau_n
    \end{cases}
\end{align}
\end{subequations}
and $\ket{\epsilon_{\alpha,\vec{\tau}}}$ are a set of distinct eigenvectors of the grain. The form~\eqref{eq:Nspin_eig} can be seen to reduce to the forms~\eqref{eq:1spin_eig} and~\eqref{eq:2spin_eig} in the cases $N=1$ and $N=2$. By generalizing the above calculation, we obtain 
\begin{equation}
    \log \left( \frac{g_{N+1}}{g_{N+1,0}} \right)  = \frac12 \sum_{n = 1}^N \eta(g_n) 
\end{equation}
as desired.

\end{document}
