feat: word follow up chat with jAI✨ (#142)

## Description
<!-- Please add PR description (don't leave blank) - example: This PR
[adds/removes/fixes/replaces] the [feature/bug/etc] -->

This pull request introduces the new jAI Follow-up Chat feature to the
jargons.dev ecosystem, enabling users to ask context-aware questions
about dictionary terms and receive intelligent, conversational AI
responses. The changes span documentation updates, codebase structure,
and a major new React component implementing the chat widget.

**Feature Addition: Interactive Follow-up Chat**
- Adds a complete React implementation for the jAI Follow-up Chat
widget, including authentication, streaming AI responses, default
question templates, and a trigger button for user interaction.

**Documentation and API Updates**
- Updates the `README.md` to describe the new follow-up chat feature,
its conversational capabilities, and context-aware Q&A, replacing
previous generic explanations.
- Documents the new `components/follow-up-chat.jsx` file and its main
components in the module structure section.
- Expands documentation for prompt templates, clarifying the distinction
between `FOLLOW_UP_CHAT` (for conversational Q&A) and `SEARCH_WORD` (for
dictionary definitions).
- Details the new API endpoint `src/pages/api/jai/follow-up-chat.js`,
including semantic search, conversation history, streaming responses,
and CORS support.
- Provides a feature breakdown for the new chat widget, including UI
elements, authentication, streaming, and integration points.

## Related Issue
<!-- Please prefix the issue number with Fixes/Resolves - example: Fixes
#123 or Resolves #123 -->

Fixes jargonsdev/roadmap#6

## Screenshots/Screencasts
<!-- Please provide screenshots or video recording that demos your
changes (especially if it's a visual change) -->

[screen-capture
(18).webm](https://github.com/user-attachments/assets/85c342b0-8682-45d4-972c-5db35d59e34e)

## Notes to Reviewer
<!-- Please state here if you added a new npm packages, or any extra
information that can help reviewer better review you changes -->

- Relies on already established `apps/jai` module
- Added new node packaages
  - @ai-sdk/react
  - js-cookies

Author: babblebey

apps/jai/README.md

@@ -13,7 +13,7 @@
 Unlike standalone AI applications, ✨jAI is deeply integrated into the jargons.dev ecosystem, powering features like:
 
 - **AI-powered word definitions**: Generate instant definitions for technical terms not yet in the dictionary
-- Intelligent word explanations and follow-up conversations
+- **Interactive follow-up chat**: Ask context-aware questions about any word and get intelligent, conversation-based explanations
 - Semantic search across the dictionary
 - Context-aware responses based on the curated dictionary content
 - Real-time AI assistance for developers exploring technical terms
@@ -34,6 +34,7 @@ The ✨jAI module is organized into focused utility files:
 apps/jai/
 ├── index.js               # Main exports and module interface
 ├── components/            # React components for UI/feature integration
+│   ├── follow-up-chat.jsx # jAI Follow-up Chat feature
 │   ├── logo.jsx           # jAI Logo
 │   └── word-search.jsx    # jAI Word Search feature
 └── lib/
@@ -55,14 +56,24 @@ export { jAIPrompts, formatMessage, model, vectorStore };
 
 #### `lib/jai-prompts.js`
 
-Defines ✨jAI's personality and conversation templates. The AI assistant is designed to:
+Defines ✨jAI's personality and conversation templates through two specialized prompts:
 
-- Explain technical jargon clearly and concisely
-- Use relatable analogies and developer-friendly examples
-- Maintain a friendly, witty personality
-- Encourage follow-up questions and deeper exploration
-- Generate accurate, SEO-friendly definitions for technical terms
-- Provide structured responses optimized for dictionary content
+**`FOLLOW_UP_CHAT`**: Powers conversational interactions where jAI:
+
+- Maintains a friendly, slightly witty personality as a knowledgeable coding buddy
+- Explains technical concepts using clear analogies and real-world comparisons
+- Provides context-aware answers based on the word being viewed
+- Encourages curiosity and deeper exploration of topics
+- Uses developer-centric examples (primarily JavaScript)
+- Maintains conversation history for coherent follow-up discussions
+
+**`SEARCH_WORD`**: Generates structured dictionary definitions that are:
+
+- Formal, clear, and beginner-friendly
+- SEO-optimized with natural keyword usage
+- Accurate and up-to-date with industry standards
+- Structured as: Meaning → Further Explanation → Example (when necessary)
+- Focused on software, programming, and technology contexts
 
 #### `lib/model.js`
 
@@ -98,6 +109,19 @@ React component that powers the AI-driven word definition feature. Includes:
 - Error handling and loading states
 - Integration with the `/api/jai/search` endpoint
 
+#### `components/follow-up-chat.jsx`
+
+Interactive chat widget component for context-aware Q&A about dictionary terms. Features:
+
+- **JAIFollowUpChatWidget**: Main chat interface with message history and streaming responses
+- **JAIWordFollowUpChatWidgetTrigger**: Floating trigger button to open the chat widget
+- Default question templates for quick interaction (e.g., "Can you give me a real-world example of {word}?")
+- Authentication state management and login prompts for non-authenticated users
+- Auto-scrolling chat container for smooth conversation flow
+- Real-time streaming of AI responses using the `@ai-sdk/react` useChat hook
+- Integration with the `/api/jai/follow-up-chat` endpoint
+- User avatar display for personalized conversations
+
 ## Environment Variables
 
 ✨jAI requires the following environment variables:
@@ -146,17 +170,28 @@ Dedicated endpoint for AI-powered word definitions that:
 - Provides fallback definitions for terms not yet in the dictionary
 - Powers the `/browse/with-jai` page and word search components
 
-<!-- ### 3. Follow-up Chat API (`src/pages/api/jai/follow-up-chat.js`)
+### 3. Follow-up Chat API (`src/pages/api/jai/follow-up-chat.js`)
+
+Interactive chat endpoint that imports all four core utilities (`jAIPrompts`, `model`, `formatMessage`, `vectorStore`) for real-time AI conversations. This powers the follow-up chat feature with:
 
-Imports all four core utilities (`jAIPrompts`, `model`, `formatMessage`, `vectorStore`) for real-time AI interactions. Powers the follow-up chat feature with semantic search for relevant context, conversation history management, and streaming AI response generation. -->
+- **Semantic Search**: Retrieves relevant dictionary content using vector similarity search to ground responses in accurate context
+- **Conversation History**: Maintains chat history for coherent, context-aware follow-up discussions
+- **Streaming Responses**: Uses LangChain's RunnableSequence to stream AI responses in real-time
+- **CORS Support**: Configured for production and preview deployments with proper cross-origin access
+- The `FOLLOW_UP_CHAT` prompt template for conversational, developer-friendly explanations
 
 ### Integration Flow
 
 1. **Data Preparation**: `seed-vector-store.js` populates the vector database with dictionary content
 2. **Word Search Flow**: Users can search for undefined terms via `/browse/with-jai`, which uses the search API to generate instant definitions
-3. **Runtime Processing**: API endpoints use ✨jAI utilities for semantic search and AI response generation
-4. **Real-time Interaction**: Streaming responses provide immediate feedback to users
-5. **Context Awareness**: Vector search ensures AI responses are grounded in dictionary content
+3. **Follow-up Chat Flow**: Users viewing dictionary words can click "Ask jAI" to open an interactive chat widget that:
+   - Offers default question templates for quick engagement
+   - Accepts custom questions about the current word or related concepts
+   - Retrieves relevant context from the vector store for accurate, grounded responses
+   - Maintains conversation history for coherent multi-turn discussions
+4. **Runtime Processing**: API endpoints use ✨jAI utilities for semantic search and AI response generation
+5. **Real-time Interaction**: Streaming responses provide immediate feedback to users
+6. **Context Awareness**: Vector search ensures AI responses are grounded in dictionary content
 
 ## Development