docs: Update project documentation and session notes

- Updates `GEMINI.md` with a detailed summary of the days debugging session for the v3 Badges page.
- Updates `TODO.md` to mark recent bug fixes as complete and adds new tasks for outstanding issues (e.g., `order_by_li` fix).
- Updates `ARCHITECTURE.md` to reflect the current state of the tech stack (Tailwind v4, component library status).
- Creates `README.md` files for the new v3 Badges module and the legacy Lead Retrieval module to clarify their purpose and status.

src/routes/events/[event_id]/(badges)/badges/README.md

@@ -0,0 +1,16 @@
+# Event Badges Module (v3)
+
+This directory contains the Svelte components and pages for the "v3" Badge Printing module for a specific event.
+
+## Key Features
+
+- **Search:** A reactive search page (`+page.svelte` and `ae_comp__badge_search.svelte`) for finding attendee badges.
+- **Printing:** Functionality for printing individual and lists of badges.
+- **CRUD Operations:** Components for creating, viewing, updating, and deleting badges.
+- **Templates:** A system for managing different badge layouts and templates.
+
+## Technical Notes
+
+- This module is built using Svelte 5 (runes) and Flowbite-Svelte for UI components.
+- It relies on the `events_func` and `ae_events__event_badge.ts` for its business logic and API communication.
+- The search functionality was recently debugged to resolve a series of build and runtime errors related to CSS conflicts, Svelte 5 reactivity, and API query parameter issues.