diff --git a/3-research-approach/approach.tex b/3-research-approach/approach.tex
index 36374bc..c6bf935 100644
--- a/3-research-approach/approach.tex
+++ b/3-research-approach/approach.tex
@@ -586,6 +586,67 @@ mode approach is sound for nuclear failures. Shows safety can be proven even
 when controller deviates from nominal (pp.85-107, UCA 1
 analysis).}
 
+\subsection{Preliminary Results}
+
+In order to demonstrate the feasibility of this research proposal, we applied
+this approach to HAHACS creation to a simple nuclear reactor model, with limited
+procedural guidance. We created a simple point kinetics reactor model with
+a two-node thermal hydraulics model. The reactor assumes plant parametrs and
+mateiral properties cosistent with a Uranium-235 pressurized water reactor. For
+teh reactor core, six precursor groups are included, and fuel temperature and
+moderator temperature feedback are implemented. For the thermal hydraulics,
+basic convection assumptions are made about the heat generated from the fuel's
+reactivity, while the coolant is assumed to be single phase water. The
+differential equations representing each part of the system are described as
+follows:
+
+\begin{equation}
+  %PKE with precursors and temperature effects
+  \label{eq:PKE}
+\end{equation}
+
+\begin{equation}
+  %thermal hydraulics
+  \label{eq:thermal_hyrdaulics}
+\end{equation}
+
+Reactivity meanwhile is a sum of several different sources. First, reactivity is
+a factor of the fuel itself. Second, temperature and fuel feedback effects will
+change the total reactivity. Finally, control rods directly add or subtract
+reactivity in the system by their movement. These equations were implemented in
+a MATLAB code to simulate the simple reactor model. 
+
+First, a discrete controller was created by writing FRETish specifications
+according to diagram shown in figure \ref{fig:hybrid_automaton}. A full list of
+the specifications created is included in the \ref{APPENDIX}.\dasnote{need to
+actually create the appendix lmao}. This was then synthesized into an automaton
+using (JKind? Kind2?)\dasnote{This will depend once reactive synthesis is done}.
+Once this automaton was created, continuous controllers were developed in MATLAB
+for each discrete control mode. The end result was a hybrid system with discrete
+modes and continuous controllers created based on logical specifications, but
+were not yet verified. 
+
+Once the continuous controllers were created, reachability analysis was
+performed on the transitory modes. For this example, there is only one
+transitory mode: the 'heatup' mode. For this mode, the entry temperature
+\(T_{MIN}\) was included as an entry condition, while other states were given
+physically reasonable bounds. Reachability analysis showed that the continuous
+controller maintened the requirement as demonstrated in
+\ref{eq:reachability}\dasnote{Need to make formal equations for earlier
+sections. That way we can reference them here easily}. 
+
+Once the transitory mode was complete, the stabilizing mode of power operation
+was analyzed using a barrier certificate search tool. % more needed here about
+%how this shit actually work. What are the bounds??
+
+Finally, the expulsory SCRAM mode was considered. For this example, the failure
+modes were kept relatively simple. We assumed possible failure modes of coolant
+fouling, where the coolant thermal properties have changed drastically, and a
+secondary system coolant leak, where power demand increases dramatically. For
+both of these cases, reachability analysis was performed to evaluate that the
+dynamics of the system could safely reach a low power low temperature state for
+the possible operating range of the reactor.
+
 \subsection{Industrial Implementation}
 
 The methodology described above must be validated on realistic systems using
@@ -612,3 +673,4 @@ system experts at Emerson ensures that implementation details of the Ovation
 platform are handled correctly. Direct industry collaboration also provides an
 immediate pathway for technology transfer and alignment with practical
 deployment requirements.
+
diff --git a/main.tex b/main.tex
index 274280e..3d73208 100644
--- a/main.tex
+++ b/main.tex
@@ -11,7 +11,7 @@
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