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Geometry of a single-neuron GeGLU
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(Write $x\!=\!(x_g,\,x_v)$, with $x_g$ the gate component and $x_v$ the value component that will ultimately flow forward.)

Scalar form:
 $f_{\text{GeGLU}}(x_g, x_v)=\text{GELU}(x_g)\;\cdot\;x_v   =x_g\,\Phi(x_g)\;x_v$

1. Domain & codomain
   • Domain: the 2-D plane $(x_g,x_v)\in\mathbb R^2$.
   • Codomain: the real line $\mathbb R$ (for one neuron) or, in the vector case, the $d_{ff}/2$-dimensional subspace spanned by the “value” projection.
   Geometrically we are mapping each point of the plane to a signed height above/below it.

2. Separating axes
   • The $x_g$-axis controls a smooth gate, ranging continuously from 0 (far negative) through 0.5 (at the origin) toward 1 (large positive).
   • The $x_v$-axis is passed through unchanged and merely gets scaled by that gate.
   Thus the plane is foliated into vertical lines (constant $x_g$) that are uniformly compressed or expanded.

3. Surface shape
   Plotting $z=f_{\text{GeGLU}}(x_g,x_v)$ gives a saddle-like surface that is:

   • Flat along the entire $x_g$ axis at $x_v\!=\!0$ (because the value is zero).
   • As $|x_v|$ increases the surface “fans out’’ in the $\pm x_v$ direction, but its amplitude is modulated by the bell-shaped factor $x_g\Phi(x_g)$ (Fig. below).

   For fixed $x_v$ the cross-section is exactly the 1-D GELU curve scaled by $x_v$; for fixed $x_g$ it is a straight line through the origin with slope $\text{GELU}(x_g)$.

4. Level sets (iso-contours)
   Solving $f(x_g,x_v)=c$ gives
   $x_v = c / \text{GELU}(x_g)$ whenever $\text{GELU}(x_g)\neq0$.
   These are rational curves that resemble hyperbolas but whose arms pinch smoothly toward the $x_g<0$ region where the gate shrinks to zero. No contour ever crosses the negative $x_g$ axis because the output there is arbitrarily close to 0.

5. Gradient field
   $\nabla f = \bigl(x_v\,\text{GELU}'(x_g),\; \text{GELU}(x_g)\bigr)$
   • Along the $x_v$ direction gradients are constant w.r.t. $x_v$: each vertical line has the same slope equal to $\text{GELU}(x_g)$.
   • Gradients vanish as $x_g\!\to\!-\infty$ (saturation), peak around $x_g\!\approx\!1$, and decay again for very large $x_g$ due to the $x_g\Phi(x_g)$ factor plateauing at $x_g$.

Vectorised GeGLU inside a Transformer FFN
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Let the input vector $h\in\mathbb R^{d}$ be linearly projected into two halves:

 $g = hW_g\in\mathbb R^{d_{ff}/2}$ (gate)
 $v = hW_v\in\mathbb R^{d_{ff}/2}$ (value)

and produce $y = \text{GELU}(g)\ \otimes\ v$. (Hadamard product.)

Geometrically:

1. The value subspace spanned by columns of $W_v$ forms a $(d_{ff}/2)$-dimensional affine subspace of the model’s representation space.
2. The gate vector $\text{GELU}(g)\in[0,\infty)^{d_{ff}/2}$ acts as a smooth, element-wise scaling mask that compresses coordinates of that subspace toward the origin depending on how “positive’’ the companion coordinates in $g$ are.
3. Consequently, the reachable set of $y$ is a warped, axis-aligned “cone’’ inside that subspace whose opening width varies smoothly with $g$. No new directions are introduced; directions are re-weighted.

Comparisons & intuition
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ReLU gate: sharp half-space clipping, output lies on two flat half-cones separated by a hard plane $x_g=0$.
GELU gate: replaces that plane by a sigmoidal transition band, bending the cone walls smoothly. The resulting surface is everywhere differentiable, giving GeGLU its smoother optimization landscape while still preserving a gating geometry (outputs shrink continuously to zero as any gate coordinate drifts negative).

Bottom line
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GeGLU carves the $(g,v)$ plane into smoothly-gated linear pencils: each “value’’ axis is a straight line whose amplitude envelope is the soft bell curve of the GELU applied to its paired “gate’’ axis. In higher dimension, this becomes an axis-aligned, differentiable gating cone embedded in the FFN’s hidden space.