From d18bf3eb01a0227c06389bbd226725ca74358d42 Mon Sep 17 00:00:00 2001
From: Luke Lowery <wyattluke.lowery@gmail.com>
Date: Mon, 3 Aug 2026 15:45:09 -0500
Subject: [PATCH 01/10] CommonMath polished docs

---
 GridKit/CommonMath.md                         | 161 ++++++++++++------
 .../Model/EMT/Component/Load/LoadZ/README.md  |   5 +-
 2 files changed, 111 insertions(+), 55 deletions(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index 74e79289e..8eff620c2 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -2,76 +2,119 @@
 
 Smooth, autodiff-friendly replacements for piecewise functions used across GridKit component models. See `CommonMath.hpp` for implementation details.
 
-## Primitives
-
-The scale $\mu=4\cdot f_{\text{sync}}=240$ is chosen so $\sigma$ behaves like a step on inputs while keeping derivatives finite. As $\mu \to \infty$, these functions approach their exact targets.
-
-| Name | Description | Usage |
-|------|-------------|-------|
-| `sigmoid` | Step function | `GENSAL`, `GENROU`, `REECA` |
-| `ramp` | Smooth one-sided ramp | `REGCA`, `REECA`, `REPCA` |
-| `qramp` | Exact one-sided quadratic ramp | `IEEET1` |
+- [Primitives](#primitives)
+- [Smooth Primitives](#smooth-primitives)
+- [Derived Functions](#derived-functions)
 
-### $\sigma$ - `sigmoid`
+## Primitives
 
-The sigmoid is also known as the logistic function. The equivalent `tanh` form is used for numerical stability because the exponential form can divide by a very large value.
+| Symbol | Name | Description |
+|--------|------|-------------|
+| $H$ | Heaviside | Unit step |
+| $\text{ReLU}$ | Rectified linear unit | One-sided ramp |
+| $\text{ReQU}$ | Rectified quadratic unit | One-sided quadratic ramp |
 
 ```math
 \begin{aligned}
-  \sigma(x)
+  H(x)
     &=
       \begin{cases}
         0 & x\le 0 \\[0pt]
         1 & x\gt 0
       \end{cases} \\[0pt]
-    &\approx \dfrac{1}{2}\left(1+\tanh\left(\dfrac{\mu x}{2}\right)\right)
+  \text{ReLU}(x)
+    &=
+    \begin{cases}
+      0 & x\le 0 \\[0pt]
+      x & x\gt 0
+    \end{cases} \\[0pt]
+  \text{ReQU}(x)
+    &=
+      \begin{cases}
+        0 & x\le 0 \\[0pt]
+        x^2 & x\gt 0
+      \end{cases} \\[0pt]
+\end{aligned}
+```
+
+## Smooth Primitives
+
+| Symbol | Name | API | Description |
+|--------|------|-----|-------------|
+| $\sigma$ | Logistic Function | `sigmoid` | Smooth Heaviside |
+| $\rho$ | Ramp | `ramp` | Smooth ReLU |
+| $q$ | Quadratic Ramp | `qramp` | Smooth ReQU |
+
+
+The default scale $\mu$ of these functions is chosen such that $\sigma'(0)=f_{\text{sync}}$, where $f_{\text{sync}}$ is the fundamental frequency of the `PhasorDynamics` model. As $\mu\to\infty$, $\rho$ and $q$ converge to $\text{ReLU}$ and $\text{ReQU}$ everywhere, while $\sigma$ converges to $H$ for $x\ne0$.
+
+### Logistic Function
+
+The logistic function is evaluated using the equivalent hyperbolic-tangent form below to avoid overflow when evaluating the exponential form directly.
+
+```math
+\begin{aligned}
+  H(x) &\approx \sigma(x) = \dfrac{1}{2}\left(1+\tanh\left(\dfrac{\mu x}{2}\right)\right)
 \end{aligned}
 ```
 
 ![](../docs/Figures/CommonMath/sigmoid.svg)
 
-### $\rho$ - `ramp`
+### Ramp
 
-`ramp` is the softplus approximation to the one-sided ramp. We do not use $x\sigma(x)$ directly because it introduces a negative tail for $x \lt 0$, while softplus stays nonnegative and approaches $\max(x, 0)$ as the smoothing becomes sharp.
+Also known as $\text{softplus}(x)$, the Ramp function is a non-negative approximation of $\text{ReLU}(x)$. We do not use $x\,\sigma(x)$ because it introduces a negative tail for $x \lt 0$, while $\text{softplus}(x)$ stays nonnegative.
 
 ```math
 \begin{aligned}
-  \rho(x)
-    &= x\,\sigma(x) \\[0pt]
-    &\approx \dfrac{x+\lvert x\rvert}{2}+\dfrac{\ln\!\left(1+e^{-\mu\lvert x\rvert}\right)}{\mu}
+  \text{ReLU}(x)
+  &\approx \rho(x) 
+  = \dfrac{1}{\mu}\ln(1+e^{\mu x})
 \end{aligned}
 ```
 
 ![](../docs/Figures/CommonMath/ramp.svg)
 
-### $q$ - `qramp`
+Although $\rho(x)$ is real-analytic, the implemented form is an overflow-safe representation of the function:
 
-*Note*: the implementation of the quadratic ramp `q(x)` could be optimized with Enzyme features down the road so that we don't need the smooth approximation.
+```math
+\begin{aligned}
+  \rho(x) = 
+  \dfrac{x+\lvert x\rvert}{2}
+  + \dfrac{1}{\mu}\ln\left(1+e^{-\mu\lvert x\rvert}\right)
+\end{aligned}
+```
+The kinks of the two terms cancel exactly.
+
+### Quadratic Ramp
+
+This approximation is implemented using the logistic function.
 
 ```math
-q(x)=x^2\,\sigma(x)
+\begin{aligned}
+  \text{ReQU}(x) &\approx q(x) = x^2\,\sigma(x)
+\end{aligned}
 ```
 
 ![](../docs/Figures/CommonMath/qramp.svg)
 
 ## Derived Functions
 
-| Name | Description | Usage |
-|------|-------------|-------|
-| `max` | Smooth binary maximum | `REGCA`, `REECA`, `REECB` |
-| `min` | Smooth binary minimum | `REGCA`, `REECA` |
-| `clamp` | Bounded saturation | `IEEEST`, `REGCA`, `REECA`, `REECB`, `REPCA` |
-| `deadband1` | Type 1 no-offset signed two-sided deadband | - |
-| `deadband2` | Type 2 offset signed two-sided deadband | `REECA`, `REECB`, `REPCA` |
-| `slew` | Symmetric slew-rate limiter | - |
-| `linseg` | Saturated linear segment contribution | `REGCA`, `REECA` |
-| `above` | Above-lower-limit indicator | `REPCA` |
-| `below` | Below-upper-limit indicator | - |
-| `inside` | Interior pulse indicator | - |
-| `outside` | Outside-band indicator | `REECA`, `REECB` |
-| `antiwindup` | Anti-windup limited derivative | `IEEET1`, `SEXS-PTI`, `TGOV1`, `REECA`, `REECB`, `REPCA` |
-
-### `max`
+| Name | API | Description |
+|------|-----|-------------|
+| Maximum | `max` | Smooth binary maximum |
+| Minimum | `min` | Smooth binary minimum |
+| Clamp | `clamp` | Bounded saturation |
+| Type 1 Deadband | `deadband1` | No-offset signed two-sided deadband |
+| Type 2 Deadband | `deadband2` | Offset signed two-sided deadband |
+| Slew | `slew` | Symmetric slew-rate limiter |
+| Linear Segment | `linseg` | Saturated linear segment contribution |
+| Above | `above` | Above-lower-limit indicator |
+| Below | `below` | Below-upper-limit indicator |
+| Inside | `inside` | Interior pulse indicator |
+| Outside | `outside` | Outside-band indicator |
+| Antiwindup | `antiwindup` | Anti-windup limited derivative |
+
+### Maximum
 
 ```math
 \begin{aligned}
@@ -87,7 +130,7 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/max.svg)
 
-### `min`
+### Minimum
 
 ```math
 \begin{aligned}
@@ -103,7 +146,9 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/min.svg)
 
-### `clamp`
+### Clamp
+
+The limits satisfy $\ell\le u$.
 
 ```math
 \begin{aligned}
@@ -120,7 +165,9 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/clamp.svg)
 
-### `deadband1`
+### Type 1 Deadband
+
+The limits satisfy $\ell\le u$.
 
 ```math
 \begin{aligned}
@@ -137,7 +184,9 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/deadband1.svg)
 
-### `deadband2`
+### Type 2 Deadband
+
+The limits satisfy $\ell\le u$.
 
 ```math
 \begin{aligned}
@@ -154,7 +203,9 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/deadband2.svg)
 
-### `slew`
+### Slew
+
+The rate limit satisfies $r\ge0$.
 
 ```math
 \begin{aligned}
@@ -171,7 +222,9 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/slew.svg)
 
-### `linseg`
+### Linear Segment
+
+The breakpoints satisfy $a<b$.
 
 ```math
 \begin{aligned}
@@ -188,7 +241,7 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/linseg.svg)
 
-### `above`
+### Above
 
 ```math
 \begin{aligned}
@@ -204,7 +257,7 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/above.svg)
 
-### `below`
+### Below
 
 ```math
 \begin{aligned}
@@ -220,15 +273,17 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/below.svg)
 
-### `inside`
+### Inside
+
+The limits satisfy $\ell\le u$.
 
 ```math
 \begin{aligned}
   \text{inside}(x;\ell,u)
     &=
       \begin{cases}
-        1 & \ell\lt x\lt u \\[0pt]
-        0 & \text{else}
+        1 & \ell\le x\le u \\[0pt]
+        0 & \text{otherwise}
       \end{cases} \\[0pt]
     &\approx \sigma(x-\ell)+\sigma(u-x)-1
 \end{aligned}
@@ -236,7 +291,9 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/inside.svg)
 
-### `outside`
+### Outside
+
+The limits satisfy $\ell\le u$.
 
 ```math
 \begin{aligned}
@@ -244,7 +301,7 @@ q(x)=x^2\,\sigma(x)
     &=
       \begin{cases}
         1 & x\lt \ell\ \lor\ x\gt u \\[0pt]
-        0 & \text{else}
+        0 & \text{otherwise}
       \end{cases} \\[0pt]
     &\approx \sigma(\ell-x)+\sigma(x-u)
 \end{aligned}
@@ -252,9 +309,9 @@ q(x)=x^2\,\sigma(x)
 
 ![](../docs/Figures/CommonMath/outside.svg)
 
-### `antiwindup`
+### Antiwindup
 
-<a id="anti-windup-indicator"></a>
+The limits satisfy $\ell\le u$.
 
 ```math
 \begin{aligned}
diff --git a/GridKit/Model/EMT/Component/Load/LoadZ/README.md b/GridKit/Model/EMT/Component/Load/LoadZ/README.md
index 14d46f0c6..15c9119af 100644
--- a/GridKit/Model/EMT/Component/Load/LoadZ/README.md
+++ b/GridKit/Model/EMT/Component/Load/LoadZ/README.md
@@ -33,9 +33,8 @@ $\mathbf{z}$ | Impedance | [VectorFit](../../../Operators/Rational/VectorFit/REA
 
 ### Submodel Validation
 
-The current is differential for a nonsingular linear coefficient and algebraic
-when the coefficient is zero. Partially singular coefficients are not
-supported.
+The current is differential when the linear coefficient has full rank and
+algebraic when it is the zero matrix. No other coefficients are supported.
 
 ```math
 \mathbf{E}^{\mathbf{z}}=\mathbf{0}

From eeb0cbdb9e5384b0685e83aa92e4a549d4a72768 Mon Sep 17 00:00:00 2001
From: lukelowry <wyattluke.lowery@gmail.com>
Date: Wed, 5 Aug 2026 21:07:35 -0500
Subject: [PATCH 02/10] Fix CommonMath trailing whitespace

---
 GridKit/CommonMath.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index 8eff620c2..f2d464ebc 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -67,7 +67,7 @@ Also known as $\text{softplus}(x)$, the Ramp function is a non-negative approxim
 ```math
 \begin{aligned}
   \text{ReLU}(x)
-  &\approx \rho(x) 
+  &\approx \rho(x)
   = \dfrac{1}{\mu}\ln(1+e^{\mu x})
 \end{aligned}
 ```
@@ -78,7 +78,7 @@ Although $\rho(x)$ is real-analytic, the implemented form is an overflow-safe re
 
 ```math
 \begin{aligned}
-  \rho(x) = 
+  \rho(x) =
   \dfrac{x+\lvert x\rvert}{2}
   + \dfrac{1}{\mu}\ln\left(1+e^{-\mu\lvert x\rvert}\right)
 \end{aligned}

From fd952d0271e7db60225d27c7eb35a6c79f150895 Mon Sep 17 00:00:00 2001
From: lukelowry <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 11:09:59 -0500
Subject: [PATCH 03/10] Minor reorganizing and rewording

---
 GridKit/CommonMath.md                         | 71 ++++++++++---------
 .../Model/EMT/Component/Load/LoadZ/README.md  |  4 +-
 2 files changed, 41 insertions(+), 34 deletions(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index f2d464ebc..48082e294 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -3,42 +3,10 @@
 Smooth, autodiff-friendly replacements for piecewise functions used across GridKit component models. See `CommonMath.hpp` for implementation details.
 
 - [Primitives](#primitives)
-- [Smooth Primitives](#smooth-primitives)
 - [Derived Functions](#derived-functions)
 
 ## Primitives
 
-| Symbol | Name | Description |
-|--------|------|-------------|
-| $H$ | Heaviside | Unit step |
-| $\text{ReLU}$ | Rectified linear unit | One-sided ramp |
-| $\text{ReQU}$ | Rectified quadratic unit | One-sided quadratic ramp |
-
-```math
-\begin{aligned}
-  H(x)
-    &=
-      \begin{cases}
-        0 & x\le 0 \\[0pt]
-        1 & x\gt 0
-      \end{cases} \\[0pt]
-  \text{ReLU}(x)
-    &=
-    \begin{cases}
-      0 & x\le 0 \\[0pt]
-      x & x\gt 0
-    \end{cases} \\[0pt]
-  \text{ReQU}(x)
-    &=
-      \begin{cases}
-        0 & x\le 0 \\[0pt]
-        x^2 & x\gt 0
-      \end{cases} \\[0pt]
-\end{aligned}
-```
-
-## Smooth Primitives
-
 | Symbol | Name | API | Description |
 |--------|------|-----|-------------|
 | $\sigma$ | Logistic Function | `sigmoid` | Smooth Heaviside |
@@ -50,6 +18,19 @@ The default scale $\mu$ of these functions is chosen such that $\sigma'(0)=f_{\t
 
 ### Logistic Function
 
+The Heaviside function $H$ is a unit step, defined:
+
+```math
+\begin{aligned}
+  H(x)
+    &=
+      \begin{cases}
+        0 & x\le 0 \\[0pt]
+        1 & x\gt 0
+      \end{cases}
+\end{aligned}
+```
+
 The logistic function is evaluated using the equivalent hyperbolic-tangent form below to avoid overflow when evaluating the exponential form directly.
 
 ```math
@@ -62,6 +43,19 @@ The logistic function is evaluated using the equivalent hyperbolic-tangent form
 
 ### Ramp
 
+The rectified linear unit $\text{ReLU}$ is a one-sided ramp:
+
+```math
+\begin{aligned}
+  \text{ReLU}(x)
+    &=
+      \begin{cases}
+        0 & x\le 0 \\[0pt]
+        x & x\gt 0
+      \end{cases}
+\end{aligned}
+```
+
 Also known as $\text{softplus}(x)$, the Ramp function is a non-negative approximation of $\text{ReLU}(x)$. We do not use $x\,\sigma(x)$ because it introduces a negative tail for $x \lt 0$, while $\text{softplus}(x)$ stays nonnegative.
 
 ```math
@@ -87,6 +81,19 @@ The kinks of the two terms cancel exactly.
 
 ### Quadratic Ramp
 
+The rectified quadratic unit $\text{ReQU}$ is a one-sided quadratic ramp:
+
+```math
+\begin{aligned}
+  \text{ReQU}(x)
+    &=
+      \begin{cases}
+        0 & x\le 0 \\[0pt]
+        x^2 & x\gt 0
+      \end{cases}
+\end{aligned}
+```
+
 This approximation is implemented using the logistic function.
 
 ```math
diff --git a/GridKit/Model/EMT/Component/Load/LoadZ/README.md b/GridKit/Model/EMT/Component/Load/LoadZ/README.md
index 15c9119af..566a4c07f 100644
--- a/GridKit/Model/EMT/Component/Load/LoadZ/README.md
+++ b/GridKit/Model/EMT/Component/Load/LoadZ/README.md
@@ -33,8 +33,8 @@ $\mathbf{z}$ | Impedance | [VectorFit](../../../Operators/Rational/VectorFit/REA
 
 ### Submodel Validation
 
-The current is differential when the linear coefficient has full rank and
-algebraic when it is the zero matrix. No other coefficients are supported.
+The current is differential when $\mathbf{E}^{\mathbf{z}}$ is nonsingular and
+algebraic when $\mathbf{E}^{\mathbf{z}}=\mathbf{0}$.
 
 ```math
 \mathbf{E}^{\mathbf{z}}=\mathbf{0}

From d0fca4513934ae0f64207ea4d9ca1a0a922c8891 Mon Sep 17 00:00:00 2001
From: Luke Lowery <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 13:06:01 -0500
Subject: [PATCH 04/10] Update GridKit/CommonMath.md

Co-authored-by: Nicholson Koukpaizan <72402802+nkoukpaizan@users.noreply.github.com>
---
 GridKit/CommonMath.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index 48082e294..d70c76f33 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -9,9 +9,9 @@ Smooth, autodiff-friendly replacements for piecewise functions used across GridK
 
 | Symbol | Name | API | Description |
 |--------|------|-----|-------------|
-| $\sigma$ | Logistic Function | `sigmoid` | Smooth Heaviside |
-| $\rho$ | Ramp | `ramp` | Smooth ReLU |
-| $q$ | Quadratic Ramp | `qramp` | Smooth ReQU |
+| $\sigma$ | Logistic Function | `sigmoid` | Smooth approximation to the Heaviside step function |
+| $\rho$ | Ramp | `ramp` | Smooth approximation to the ramp function |
+| $q$ | Quadratic Ramp | `qramp` | Smooth approximation to the quadratic ramp function |
 
 
 The default scale $\mu$ of these functions is chosen such that $\sigma'(0)=f_{\text{sync}}$, where $f_{\text{sync}}$ is the fundamental frequency of the `PhasorDynamics` model. As $\mu\to\infty$, $\rho$ and $q$ converge to $\text{ReLU}$ and $\text{ReQU}$ everywhere, while $\sigma$ converges to $H$ for $x\ne0$.

From 77bd848ae9cb5eb7c8ef73950f1c9bd506ea946c Mon Sep 17 00:00:00 2001
From: Luke Lowery <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 13:06:47 -0500
Subject: [PATCH 05/10] Update GridKit/CommonMath.md

Co-authored-by: Nicholson Koukpaizan <72402802+nkoukpaizan@users.noreply.github.com>
---
 GridKit/CommonMath.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index d70c76f33..c3ddf1f73 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -18,7 +18,7 @@ The default scale $\mu$ of these functions is chosen such that $\sigma'(0)=f_{\t
 
 ### Logistic Function
 
-The Heaviside function $H$ is a unit step, defined:
+The Heaviside function $H$ is a unit step, defined as:
 
 ```math
 \begin{aligned}

From 5f4db812f73bde34f76733d76a6f7d8c4f5dc68b Mon Sep 17 00:00:00 2001
From: Luke Lowery <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 13:07:17 -0500
Subject: [PATCH 06/10] Update GridKit/CommonMath.md

Co-authored-by: Nicholson Koukpaizan <72402802+nkoukpaizan@users.noreply.github.com>
---
 GridKit/CommonMath.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index c3ddf1f73..90a45fe69 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -31,7 +31,7 @@ The Heaviside function $H$ is a unit step, defined as:
 \end{aligned}
 ```
 
-The logistic function is evaluated using the equivalent hyperbolic-tangent form below to avoid overflow when evaluating the exponential form directly.
+We use the logistic function as a smooth approximation to the Heaviside function. This approximation is evaluated using the equivalent hyperbolic-tangent form below to avoid overflow when evaluating the exponential form directly.
 
 ```math
 \begin{aligned}

From 1864eb66600404168957827633e0241a8533eeac Mon Sep 17 00:00:00 2001
From: Luke Lowery <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 13:07:38 -0500
Subject: [PATCH 07/10] Update GridKit/CommonMath.md

Co-authored-by: Nicholson Koukpaizan <72402802+nkoukpaizan@users.noreply.github.com>
---
 GridKit/CommonMath.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index 90a45fe69..7ba68c101 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -35,7 +35,7 @@ We use the logistic function as a smooth approximation to the Heaviside function
 
 ```math
 \begin{aligned}
-  H(x) &\approx \sigma(x) = \dfrac{1}{2}\left(1+\tanh\left(\dfrac{\mu x}{2}\right)\right)
+  \sigma(x) = \dfrac{1}{2}\left(1+\tanh\left(\dfrac{\mu x}{2}\right)\right) &\approx H(x)  
 \end{aligned}
 ```
 

From 0d0cc6c17b56e9c4fa104d9f0c75dd198aa3b005 Mon Sep 17 00:00:00 2001
From: lukelowry <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 15:57:38 -0500
Subject: [PATCH 08/10] clarify activation fucntion application

---
 GridKit/CommonMath.md | 12 +++++++-----
 1 file changed, 7 insertions(+), 5 deletions(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index 7ba68c101..32bdd06a8 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -1,6 +1,9 @@
 # CommonMath
 
-Smooth, autodiff-friendly replacements for piecewise functions used across GridKit component models. See `CommonMath.hpp` for implementation details.
+Smooth, autodiff-friendly approximations of piecewise functions used across
+GridKit component models. In model equations, these functions act as smooth
+activation functions for saturation, limits, deadbands, and antiwindup behavior. See
+`CommonMath.hpp` for implementation details.
 
 - [Primitives](#primitives)
 - [Derived Functions](#derived-functions)
@@ -60,9 +63,8 @@ Also known as $\text{softplus}(x)$, the Ramp function is a non-negative approxim
 
 ```math
 \begin{aligned}
-  \text{ReLU}(x)
-  &\approx \rho(x)
-  = \dfrac{1}{\mu}\ln(1+e^{\mu x})
+  \rho(x)
+  = \dfrac{1}{\mu}\ln(1+e^{\mu x}) \approx \text{ReLU}(x)
 \end{aligned}
 ```
 
@@ -98,7 +100,7 @@ This approximation is implemented using the logistic function.
 
 ```math
 \begin{aligned}
-  \text{ReQU}(x) &\approx q(x) = x^2\,\sigma(x)
+  q(x) = x^2\,\sigma(x) \approx \text{ReQU}(x) 
 \end{aligned}
 ```
 

From d55036ae23b025dd2983429e908ceef33d7d2911 Mon Sep 17 00:00:00 2001
From: Luke Lowery <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 18:40:24 -0500
Subject: [PATCH 09/10] Update GridKit/CommonMath.md

Co-authored-by: Nicholson Koukpaizan <72402802+nkoukpaizan@users.noreply.github.com>
---
 GridKit/CommonMath.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index 32bdd06a8..cf7e74ba6 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -59,7 +59,7 @@ The rectified linear unit $\text{ReLU}$ is a one-sided ramp:
 \end{aligned}
 ```
 
-Also known as $\text{softplus}(x)$, the Ramp function is a non-negative approximation of $\text{ReLU}(x)$. We do not use $x\,\sigma(x)$ because it introduces a negative tail for $x \lt 0$, while $\text{softplus}(x)$ stays nonnegative.
+We implement the Ramp function as the $\text{softplus}(x)$, non-negative approximation of $\text{ReLU}(x)$. We do not use $x\,\sigma(x)$ because it introduces a negative tail for $x \lt 0$, while $\text{softplus}(x)$ stays nonnegative.
 
 ```math
 \begin{aligned}

From 58c9488b91aa5e2bba10f19d32d972cf65ba01d5 Mon Sep 17 00:00:00 2001
From: Luke Lowery <wyattluke.lowery@gmail.com>
Date: Thu, 6 Aug 2026 18:40:45 -0500
Subject: [PATCH 10/10] Update GridKit/CommonMath.md

Co-authored-by: Nicholson Koukpaizan <72402802+nkoukpaizan@users.noreply.github.com>
---
 GridKit/CommonMath.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/GridKit/CommonMath.md b/GridKit/CommonMath.md
index cf7e74ba6..d8f409c85 100644
--- a/GridKit/CommonMath.md
+++ b/GridKit/CommonMath.md
@@ -96,7 +96,7 @@ The rectified quadratic unit $\text{ReQU}$ is a one-sided quadratic ramp:
 \end{aligned}
 ```
 
-This approximation is implemented using the logistic function.
+We implement an approximation to $\text{ReQU}$ using the logistic function.
 
 ```math
 \begin{aligned}