Begin by visiting the Twitter Developer Portal and creating a developer account if you haven't already. Once set up, create a new application. This application will allow you to access Twitter's API. Note down the API key, API secret key, Access token, and Access token secret that are generated for your app. These credentials are essential for authenticating your requests to Twitter's API.

Make sure you have Python installed on your system. You'll need to install the 'Tweepy' library, which is a Python wrapper for the Twitter API. Use the following command to install it:
```bash
pip install tweepy
```
This package will help you interact with the Twitter API programmatically.

Create a Python script to authenticate with Twitter using Tweepy. Initiate a connection to the Twitter API with your credentials:
```python
import tweepy

consumer_key = 'your_api_key'
consumer_secret = 'your_api_secret_key'
access_token = 'your_access_token'
access_token_secret = 'your_access_token_secret'

auth = tweepy.OAuth1UserHandler(consumer_key, consumer_secret, access_token, access_token_secret)
api = tweepy.API(auth)
```
This setup ensures that your app can securely communicate with Twitter's API.

Use the authenticated API object to fetch data from Twitter. For example, to get tweets from a specific user:
```python
tweets = api.user_timeline(screen_name='twitter_user', count=100, tweet_mode='extended')
```
Adjust parameters as necessary to fetch the specific data you need, such as tweets from a user's timeline, search queries, etc.

Once you have the data, transform it into JSON format. You can use Python's built-in `json` library to convert Tweepy's response objects into a JSON-serializable format:
```python
import json

tweets_json = [tweet._json for tweet in tweets]
```
This step ensures that the data structure is ready for JSON serialization.

Create a JSON file and write the transformed data into it. This step involves opening a file in write mode and dumping the JSON data into it:
```python
with open('tweets.json', 'w') as json_file:
json.dump(tweets_json, json_file, indent=4)
```
The `indent=4` argument formats the JSON data to be more readable.

After writing the data to the JSON file, verify its integrity by reading the file and printing a few entries to ensure everything is written correctly:
```python
with open('tweets.json', 'r') as json_file:
data = json.load(json_file)
print(data[:5]) # Print the first 5 tweets
```
This step confirms that your data export process was successful and reliable.

By following these steps, you will be able to move data from Twitter to a JSON file without using third-party connectors or integrations.