From 205ea85a65e8da79f79ac9018a95366feba0ac84 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 6 Jun 2023 09:43:11 +0200
Subject: Some writing.

---
 2-point.tex | 35 ++++++++++++++++++++---------------
 1 file changed, 20 insertions(+), 15 deletions(-)

diff --git a/2-point.tex b/2-point.tex
index bf1a5ca..efc7a71 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -670,28 +670,33 @@ Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case,
   \tilde r_\mathrm d^{00}=r^{00}_\mathrm df'(1) &&
   \tilde d_\mathrm d^{00}=d^{00}_\mathrm df'(1)
 \end{align}
-
-
+Once these expressions are inserted into the complexity, the limits of $n$ and
+$m$ to zero can be taken, and the parameters from $D^{01}$ and $D^{11}$ can be
+extremized explicitly. The resulting expression for the complexity, which must
+still be extremized over the parameters $\hat\beta_1$, $r^{01}$,
+$r^{11}_\mathrm d$, $r^{11}_0$, and $q^{11}_0$, is
 \begin{equation}
   \begin{aligned}
-    &\Sigma_{12}=\mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1
-    +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)
-    +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q)
-    \\
-    &+\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)+
-    \frac12\Bigg\{
+    &\Sigma_{12}(E_0,\mu_0,E_1,\mu_1,q)
+    =\mathop{\mathrm{extremum}}_{\hat\beta_1,r^{11}_\mathrm d,r^{11}_0,r^{01},q^{11}_0}\Bigg\{
+    \mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1
+    +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)\\
+    &\qquad+\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q)
+    +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)\\
+    &+\frac12\Bigg[
       \hat\beta_1^2\big(f(1)-f(q^{11}_{0})\big)
       +(r^{11}_\mathrm d)^2f''(1)+2r^{01}r^{10}f''(q)-(r^{11}_0)^2f''(q^{11}_0)
+      +\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1)
       \\
-    &+\left(
-      (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right)
+    &\qquad+\frac{1-q^2}{1-q^{11}_0}+\left(
+      (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}
+      \left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right)
     \right)\big(f'(1)-f'(q_{22}^{(0)})\big) \\
-    &+\frac{1-q^2}{1-q^{11}_0}+\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1)
+    &\qquad
     -\frac1{f'(1)}\frac{f'(1)^2-f'(q)^2}{f'(1)-f'(q^{11}_0)}
     +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)
-    \\
-    &+\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right)
-  \Bigg\}
+    +\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right)
+  \Bigg]\Bigg\}
   \end{aligned}
 \end{equation}
 
@@ -713,7 +718,7 @@ expanding in powers of $\Delta q=1-q$. For the complexity, the result is
   \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q)
   +O\big((1-q)^2\big)
 \end{equation}
-The popular of stationary points that are most common at each energy have the relation
+The population of stationary points that are most common at each energy have the relation
 \begin{equation}
   E_\mathrm{dom}(\mu_0)=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0
 \end{equation}
-- 
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