The user wants a single-page web app that replicates the "Matrix digital rain" effect.
Key requirements:
1. Full-screen black canvas.
2. Columns of falling green Katakana glyphs.
3. Trailing fade effect.
4. Brighter leading character.
5. All in one `index.html` file.
6. No external resources (no CDNs, no network requests).

Plan:
1. Create an `index.html` file.
2. Add a `<canvas>` element that fills the screen.
3. Add CSS to make the canvas full-screen and the background black.
4. Add JavaScript to:
    - Initialize the canvas and handle window resizing.
    - Define Katakana characters (or a range of Unicode characters).
    - Implement the animation loop:
        - Draw a semi-transparent black rectangle over the whole canvas to create the trailing effect.
        - Iterate through columns.
        - Draw characters at their respective positions.
        - Update the vertical positions of the characters.
        - Handle the "leading character" being brighter.

Let's refine the Katakana part. I'll need a string of Katakana characters.
Unicode for Katakana is roughly `\u30A0` to `\u30FF`.

Wait, the prompt says "no external resources". I can use standard Unicode characters in a string.

Implementation details:
- `canvas.getContext('2d')`
- `requestAnimationFrame` for the loop.
- An array of objects to keep track of each column's state (y-position, speed, etc.).
- The "trailing fade" is achieved by drawing `ctx.fillStyle = 'rgba(0, 0, 0, 0.05)'; ctx.fillRect(0, 0, width, height);` every frame.
- The "leading character" can be drawn with a different color or brightness.

Let's draft the code.

The code seems okay, but let's test it (mentally or by reviewing the logic).
The "head" is white (`#fff`), and the "trail" is formed by the semi-transparent black rectangle `rgba(0, 0, 0, 0.05)`.
As the white character is drawn, it stays on the screen. In subsequent frames, it gets covered by the semi-transparent black, making it look like it's fading from white to green to dark green to black.
Actually, if I draw `#fff`, it will fade to grey/black.
If I want it to fade from bright green to dark green, I should draw the head in a bright green, and the trail will also be green.
Wait, if the background is black, and I draw `#fff`, it will fade to dark grey.
If I want the trail to be green, the characters themselves must be green.

Let's refine the drawing logic:
1. Draw `rgba(0, 0, 0, 0.05)` over everything. This darkens everything.
2. Draw the character at the current position.
3. If we want the trail to be green, the character we draw MUST be green.
4. But if we want a "brighter leading character", we can draw it in white or a very bright green.
5. As it's covered by the black rect, it will fade.

If I draw white, it fades to black (through grey).
If I draw bright green, it fades to black (through dark green).
The user said: "columns of falling green katakana glyphs with a trailing fade and a brighter leading character".

So:
- Trail = Green.
- Head = Brighter (maybe White or Light Green).

Let's try this:
Every frame:
1. `ctx.fillStyle = 'rgba(0, 0, 0, 0.05)'; ctx.fillRect(0, 0, width, height);`
2. For each drop:
   `ctx.fillStyle = '#0F0';` (Green)
   `ctx.fillText(char, x, y);`
   Wait, if I only draw the head, the previous heads will fade.
   If I draw the head in green, it will fade to black.
   If I draw the head in bright green, it will fade to dark green, then black.
   
Actually, if I draw the head as `#fff` and the rest as `#0F0`, it won't work because the "rest" are just the old heads that have been faded.

Let's reconsider:
The "trailing fade" is an illusion created by NOT clearing the canvas fully.
If I draw a character at `(x, y)` in frame 1, and then in frame 2 I draw a semi-transparent black rect, the character at `(x, y)` becomes darker.
If the character was Green, it becomes Darker Green.
If the character was White, it becomes Grey.

To get Green trail with a White head:
The head is drawn at `y`. The character at `y-1` was drawn in the previous frame.
If I want the character at `y-1` to be green, I should have drawn it green.
But if I want the character at `y` to be white, I draw it white.
In the next frame, the white character at `y` will be covered by `rgba(0, 0, 0, 0.05)`, so it becomes slightly grey.
If I want it to become green, that's harder.

Actually, most Matrix implementations just draw the head in white/bright green and the rest of the trail is just the previous characters which were also bright green and have been faded.

Let's try this:
In each frame, for each drop:
1. Draw the head character in white (`#fff`).
2. This head character will fade over time because of the `rgba(0, 0, 0, 0.05)` rect.
3. Wait, if it's white, it fades to black. If I want it to be green, I should draw it in green.
4. BUT, if I want a "brighter" head, I can draw it in white, and the fade will make it dark.
5. What if I draw the head in `#fff` and the trailing characters are already there (from previous frames) but they were drawn in green? No, that doesn't work with the "fade" technique because the fade is global.

Let's use the standard approach:
The "trail" is just the result of the `rgba(0, 0, 0, 0.1)` fill.
Every frame, we draw the new character at the head of the drop.
If we draw the head in white, it will fade from white -> grey -> black.
If we draw the head in green, it will fade from green -> dark green -> black.

To get "Green trail with a brighter leading character":
We can draw the head in white, and then we can't easily make it green later unless we draw more than just the head.
But wait! If we draw the head in white, it stays white for a bit (as it's covered by 0.05 opacity) and then becomes dark.
If we want the trail to be green, we can draw a green character *behind* the white one? No.

Actually, here is a trick:
In each frame, for each drop:
1. Draw the head character in white.
2. BUT, we also need the green trail.
The "trail" is actually just the previous characters. If we want them to be green, we should have drawn them green.

Let's try this:
For each drop, instead of just one character, we draw a sequence? No, that's expensive.

What if we do this:
1. Draw the semi-transparent black rect.
2. For each drop:
   - Pick a random char.
   - Draw it in green.
   - If it's the "head" (which it always is in this simple model), draw it in white.

Wait, if I draw the character in white, it is the only thing I draw for that drop in that frame.
In the next frame, the black rect makes it darker.
If it was white, it's now dark grey.
To make it green, I'd have to draw green over it.

Actually, many Matrix effects don't use a single "head". They just use different colors for characters.
But the user specifically asked for: "columns of falling green katakana glyphs with a trailing fade and a brighter leading character".

Let's look at how others do it.
A common way is to have the `drops` array store the `y` position.
In each frame:
- Fill with `rgba(0, 0, 0, 0.1)`.
- For each drop:
  - `ctx.fillStyle = '#0F0'`
  - `ctx.fillText(char, x, y)`
  - `drops[i]++`
  - To get a bright head, we can draw another `fillText` with `#fff` at the same position? No, that's the same position.

Wait! If I draw the head in white, it will fade. If I want it to fade *through* green, I can't easily do that with the global fade.
*Unless* I don't use a global fade for the head? No, that's too complex.

Actually, if I draw the head in `#fff`, it will fade to black.
If I draw the character in `#0F0`, it will fade to black.
If I want a green trail with a white head, I can draw the character in green, and then *also* draw it in white? No, that's just white.

How about this:
The "trail" is actually the characters that were drawn in previous frames.
If I draw the character in Green, it will fade to dark green.
If I draw the character in White, it will fade to dark grey.

Wait, I have an idea.
The "trail" isn't just the character. It's the fact that the character stays there.
If I draw a character in white at `y`, and in the next frame I draw a character in green at `y + fontSize`, the white one is still at `y`.
If I want the white one to eventually become green... that's not how it works.

Let's try the most visual way:
1. Head is white.
2. The trail is green.
To achieve this, I can draw the character in green every frame, but the *current* position of the head gets an extra white character drawn on top of it.
BUT, the white character will also be faded by the black rect.
So it will go: White -> Light Grey -> Dark Grey -> Black.
The green character will go: Green -> Dark Green -> Black.
This actually looks pretty good! The "head" is white, and as it moves down, it becomes darker.
If I want it to turn green, I can't really do that with the global fade.
But if the "trail" is green, it means the characters *behind* the head are green.
So:
- At `y` (head): White character.
- At `y - fontSize`, `y - 2*fontSize`, etc.: Green characters.

But wait, if I only draw the head at `y` in each frame, then the character at `y - fontSize` is just the one I drew in the *previous* frame.
So:
- Frame 1: Draw white char at `y`.
- Frame 2: Draw black rect. Draw white char at `y + fontSize`.
Now we have a white char at `y` (faded) and a white char at `y + fontSize`.
This creates a white trail.

To get a green trail, I need to draw green characters.
If I draw a green char at `y - fontSize` every frame? No.

Actually, the simplest way to get a green trail with a white head:
In each frame:
1. `ctx.fillStyle = 'rgba(0, 0, 0, 0.05)'; ctx.fillRect(0, 0, width, height);`
2. For each drop:
   - `ctx.fillStyle = '#0F0';`
   - `ctx.fillText(char, x, y - fontSize);` // Draw a green character slightly behind the head? No.
   
Let's try this logic:
In each frame, for each drop:
1. We only draw the head.
2. The head is drawn in white.
3. To make the trail green, we can't use the global fade alone if we want the head to *become* the trail.

Wait! I can draw the head in white, AND I can draw the character at `y - fontSize` in green.
But that's still not quite right.

Let's look at the requirement again: "columns of falling green katakana glyphs with a trailing fade and a brighter leading character".
This often means the "glow" or "head" is white.
If I draw the head in white and the rest of the column in green...
But the column is just the history of the head.

Let's try this:
1. `ctx.fillStyle = 'rgba(0, 0, 0, 0.1)'; ctx.fillRect(0, 0, width, height);`
2. For each drop:
   - `ctx.fillStyle = '#0F0';`
   - `ctx.fillText(char, x, y);`
   - Then, to make the head brighter:
   - `ctx.fillStyle = '#fff';`
   - `ctx.fillText(char, x, y);`
This is still just white.

Wait, if I draw the character in green, and then I draw it in white, it's just white.
UNLESS I draw the green one at `y` and the white one at `y - fontSize`? No.

How about this:
The "trail" is the characters we drew in the past.
If we want them to be green, they must have been drawn green.
If we want the current one to be white, it must be drawn white.
So, in each frame:
- Draw the head at `y` in white.
- Draw the character at `y - fontSize` in green.
- This way, the character at `y` is white. In the next frame, it's at `y + fontSize` and it's white.
Wait, that's still a white trail.

Let's try a different approach for the trail.
What if we don't use a global fade? No, that's the standard way.

Let's try this:
The `drops` array stores the current `y`.
In each frame:
1. `ctx.fillStyle = 'rgba(0, 0, 0, 0.1)'; ctx.fillRect(0, 0, width, height);`
2. For each drop:
   - `ctx.fillStyle = '#0F0';`
   - `ctx.fillText(char, x, y);`
   - `ctx.fillStyle = '#fff';`
   - `ctx.fillText(char, x, y);`
Still just white.

Wait, I'm overthinking. If I draw the character in white, it will fade. If I want a green trail, I can just draw a green character at the position *just before* the head.
But the "trail" is many characters long.

Actually, the most common way to do this is:
The "head" is a bright color. The "trail" is the same color but it's fading.
If the user wants a *green* trail and a *brighter* head, they probably mean the head is white or light green, and the trail is green.

If I draw the head in white, it fades to grey.
If I draw the head in green, it fades to dark green.
If I want the head to be white and the trail to be green, I can't easily use the global fade.

*However*, I can draw the head in white, and then *separately* draw green characters at the positions where the head *was*.
But how do I know where the head was? I can store the history of positions for each drop.
That might be too much.

Actually, there's a much simpler way.
In each frame, for each drop:
1. `ctx.fillStyle = 'rgba(0, 0, 0, 0.1)'; ctx.fillRect(0, 0, width, height);`
2. `ctx.fillStyle = '#0F0';`
3. `ctx.fillText(char, x, y);`
4. `ctx.fillStyle = '#fff';`
5. `ctx.fillText(char, x, y - fontSize);` // Draw a white character one step behind the head.

Let's test this idea.
Frame 1: Head at `y` is White. Head at `y-1` is Green.
Frame 2: Head at `y+1` is White. Head at `y` is Green.
The character at `y` was White in Frame 1, and now it's Green in Frame 2.
The black rect will fade the White character at `y` towards black.
Then the Green character at `y` will be drawn on top of it!
This is perfect! The white head will leave a green trail because in the next frame we draw a green character at its previous position.

Wait, `ctx.fillText(char, x, y - fontSize)` will draw a green character at the *previous* position.
But it's the *same* character? The user might want different characters in the trail.
Yes, I should probably pick a random character for the green one too.

Let's refine:
For each drop:
- `const headChar = charArray[Math.floor(Math.random() * charArray.length)];`
- `const trailChar = charArray[Math.floor(Math.random() * charArray.length)];`
- `ctx.fillStyle = '#0F0';`
- `ctx.fillText(trailChar, x, y - fontSize);`
- `ctx.fillStyle = '#fff';`
- `ctx.fillText(headChar, x, y);`

But wait, if I do this, the white character at `y` in Frame 1 becomes a green character at `y` in Frame 2.
The black rect will fade the white character.
This sounds like it would work and look great!

Let's check the coordinates.
If `y` is the current head position.
`y - fontSize` is the previous position.
In Frame 1, we draw white at `y`.
In Frame 2, we draw black rect (fading white at `y`).
Then we draw green at `y` (the previous `y`) and white at `y + fontSize`.
So the white character at `y` is now covered by a green character at `y`.
This is exactly what we want!

Wait, `y` is usually increasing. So `y - fontSize` is the previous position.
In Frame 1: `y = 100`. Draw White at 100.
In Frame 2: `y = 116`.
  1. Draw Black Rect (fades White at 100).
  2. Draw Green at `116 - 16 = 100`.
  3. Draw White at `116`.
Now at `100`, we have a green character.
This works!

Let's try to implement this. I will also use `ctx.font = fontSize + 'px monospace'` and ensure it's set.

One detail: the "trail" in the Matrix is often quite long. The `rgba(0, 0, 0, 0.05)` handles the "length" of the trail by how long it takes for a character to disappear.

Let's write the final code.

Wait, I just realized something about my "head/trail" logic.
If I draw `trailChar` at `y - fontSize` and `headChar` at `y` in every frame, then in every frame, I'm actually drawing *two* characters for each drop.
One is the "new" head (white) and one is the "new" trail (green).
But the "trail" is also the *previous* head.
If I draw a green character at `y - fontSize` (which was the previous head position) and a white character at `y`, then the white character at `y` will be drawn, and in the next frame, it will be at `y + fontSize` and a green character will be drawn at `y`.
This actually means that the "head" is always a white character, and the character immediately behind it is a green character.
But what about the characters *behind* that? They are just the result of the previous frames' green characters being faded by the black rect.

So the sequence of colors for a single drop at a fixed $x$ would be:
Frame $t$: White at $y_t$, Green at $y_t - \Delta y$.
Frame $t+1$: White at $y_{t+1}$, Green at $y_{t+1} - \Delta y$.
Note that $y_t = y_{t+1} - \Delta y$.
So at position $y_t$, in frame $t$ we had White, and in frame $t+1$ we have Green.
In frame $t+2$, the position $y_t$ will be covered by the black rect again, and we won't draw anything at $y_t$ (unless it's the new $y_{t+2} - \Delta y$ which it isn't).
So the color at $y_t$ goes: White $\to$ (Black rect) $\to$ Green $\to$ (Black rect) $\to$ Dark Green $\to$ (Black rect) $\to$ Darker Green $\dots$ $\to$ Black.

This is actually quite clever and produces exactly what was requested!
One small thing: the "trail" character might be different from the "head" character, which is good for variety.

Let's double check the `y - fontSize` logic.
If `y` is the current position of the head.
The head is at `y`.
The character at `y - fontSize` is the one that was at `y` in the previous frame.
In the previous frame, we drew a White character at `y`.
In this frame, we draw a Green character at `y - fontSize` (which is the old `y`).
So the color at the old `y` changes from White to Green.
This is perfect.

One potential issue: if `drops[i]` is small (near 0), `y - fontSize` could be negative. `fillText` handles negative coordinates fine, it just draws off-screen.

Let's verify the Katakana range.
`ｱｲｳｴｵｶｷｸｹｺｻｼｽｾｿﾀﾁﾂﾃﾄﾅﾆﾇﾈﾉﾊﾋﾌﾍﾎﾏﾐﾑﾒﾓﾔﾕﾖﾗﾘﾙﾚﾛﾜﾝ` is indeed Katakana.

I'll do a quick check on the visual appearance. The "trail" might look a bit "jumpy" if the green character is different from the previous white one, but that's actually how the Matrix effect looks (it's a stream of random characters).

Wait, one more thing. The user asked for "a trailing fade". My current implementation has the white head and the green trail.
If the `rgba(0, 0, 0, 0.08)` is too fast or too slow, I can adjust it.
`0.08` seems like a good middle ground.

Let's refine the character set to be more complete if possible, but the current one is already quite good.

I'll also add a `title` and some basic meta tags.

One minor improvement: the `drops[i]` can be a float to allow for smoother motion if I wanted, but using `fontSize` increments is standard for this effect to keep characters aligned to a grid.

Let's try to run it? I can't "run" it in the sense of seeing it, but I can ensure it's a valid HTML file.

Wait, I noticed that `y - fontSize` might cause the green character to be drawn at the *exact* same position as the previous white character.
If `y` was `100` and `fontSize` is `18`.
Frame 1: `y=100`. Draw White at 100.
Frame 2: `y=118`. Draw Green at `118 - 18 = 100`. Draw White at 118.
Yes, it works.

I'll make one change: I'll make the characters a bit more "matrix-y" by adding more characters and maybe adjusting the font size slightly.

Actually, the user asked for "falling green katakana glyphs". I should make sure the Katakana are the primary characters.

Let's check if I should use `ctx.globalCompositeOperation`? No, the current method is simpler and works.

Final check of the requirements:
- Full-screen black canvas: Yes.
- Columns of falling green katakana glyphs: Yes.
- Trailing fade: Yes.
- Brighter leading character: Yes.
- Single index.html file: Yes.
- No external resources: Yes.

One thing to consider: `window.innerHeight` and `window.innerWidth` might not be updated if the window is resized, but I have the `resize` event listener.

I'll add a little more variety to the Katakana.

```javascript
const chars = 'ｱｲｳｴｵｶｷｸｹｺｻｼｽｾｿﾀﾁﾂﾃﾄﾅﾆﾇﾈﾉﾊﾋﾌﾍﾎﾏﾐﾑﾒﾓﾔﾕﾖﾗﾘﾙﾚﾛﾜﾝ';
```

Actually, I'll use a larger set of characters.

One more thing: the user said "a brighter leading character". In my current code, the leading character is `#FFF` (White). This is very bright.

Let's finalize the code.