feat(#384): Add Synapse + Element Web to docker-compose for dev

- Create docker-compose.matrix.yml as optional dev overlay
- Add Synapse homeserver config with shared PostgreSQL
- Add Element Web client config (port 8501)
- Add bot account setup script (docker/matrix/scripts/setup-bot.sh)
- Add Makefile targets: matrix-up, matrix-down, matrix-logs, matrix-setup-bot
- Document Matrix env vars in .env.example
- Synapse accessible at localhost:8008, Element at localhost:8501
- Usage: docker compose -f docker/docker-compose.yml -f docker/docker-compose.matrix.yml up

Refs #384

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

docker/docker-compose.matrix.yml

@@ -0,0 +1,129 @@
+# ==============================================
+# Matrix Dev Environment (Synapse + Element Web)
+# ==============================================
+#
+# Development-only overlay for testing the Matrix bridge locally.
+# NOT for production — use docker-compose.sample.matrix.yml for production.
+#
+# Usage:
+#   docker compose -f docker/docker-compose.yml -f docker/docker-compose.matrix.yml up -d
+#
+# Or with Makefile:
+#   make matrix-up
+#
+# This overlay:
+#   - Adds Synapse homeserver (localhost:8008) using shared PostgreSQL
+#   - Adds Element Web client (localhost:8501)
+#   - Creates a separate 'synapse' database in the shared PostgreSQL instance
+#   - Enables open registration for easy dev testing
+#
+# After first startup, create the bot account:
+#   docker/matrix/scripts/setup-bot.sh
+#
+# ==============================================
+
+services:
+  # ======================
+  # Synapse Database Init
+  # ======================
+  # Creates the 'synapse' database and user in the shared PostgreSQL instance.
+  # Runs once and exits — idempotent, safe to run repeatedly.
+  synapse-db-init:
+    image: postgres:17-alpine
+    container_name: mosaic-synapse-db-init
+    restart: "no"
+    environment:
+      PGHOST: postgres
+      PGPORT: 5432
+      PGUSER: ${POSTGRES_USER:-mosaic}
+      PGPASSWORD: ${POSTGRES_PASSWORD:-mosaic_dev_password}
+      SYNAPSE_DB: ${SYNAPSE_POSTGRES_DB:-synapse}
+      SYNAPSE_USER: ${SYNAPSE_POSTGRES_USER:-synapse}
+      SYNAPSE_PASSWORD: ${SYNAPSE_POSTGRES_PASSWORD:-synapse_dev_password}
+    entrypoint: ["sh", "-c"]
+    command:
+      - |
+        until pg_isready -h postgres -p 5432 -U $${PGUSER}; do
+          echo "Waiting for PostgreSQL..."
+          sleep 2
+        done
+        echo "PostgreSQL is ready. Creating Synapse database and user..."
+
+        psql -h postgres -U $${PGUSER} -tc "SELECT 1 FROM pg_roles WHERE rolname='$${SYNAPSE_USER}'" | grep -q 1 || \
+          psql -h postgres -U $${PGUSER} -c "CREATE USER $${SYNAPSE_USER} WITH PASSWORD '$${SYNAPSE_PASSWORD}';"
+
+        psql -h postgres -U $${PGUSER} -tc "SELECT 1 FROM pg_database WHERE datname='$${SYNAPSE_DB}'" | grep -q 1 || \
+          psql -h postgres -U $${PGUSER} -c "CREATE DATABASE $${SYNAPSE_DB} OWNER $${SYNAPSE_USER} ENCODING 'UTF8' LC_COLLATE='C' LC_CTYPE='C' TEMPLATE template0;"
+
+        echo "Synapse database ready: $${SYNAPSE_DB}"
+    depends_on:
+      postgres:
+        condition: service_healthy
+    networks:
+      - mosaic-network
+
+  # ======================
+  # Synapse (Matrix Homeserver)
+  # ======================
+  synapse:
+    image: matrixdotorg/synapse:latest
+    container_name: mosaic-synapse
+    restart: unless-stopped
+    environment:
+      SYNAPSE_CONFIG_DIR: /data
+      SYNAPSE_CONFIG_PATH: /data/homeserver.yaml
+    ports:
+      - "${SYNAPSE_CLIENT_PORT:-8008}:8008"
+      - "${SYNAPSE_FEDERATION_PORT:-8448}:8448"
+    volumes:
+      - ./matrix/synapse/homeserver.yaml:/data/homeserver.yaml:ro
+      - synapse_data:/data/media_store
+      - synapse_signing_key:/data/keys
+    depends_on:
+      postgres:
+        condition: service_healthy
+      synapse-db-init:
+        condition: service_completed_successfully
+    healthcheck:
+      test: ["CMD-SHELL", "curl -fSs http://localhost:8008/health || exit 1"]
+      interval: 15s
+      timeout: 5s
+      retries: 5
+      start_period: 30s
+    networks:
+      - mosaic-network
+    labels:
+      com.mosaic.service: "matrix-synapse"
+      com.mosaic.description: "Matrix homeserver (dev)"
+
+  # ======================
+  # Element Web (Matrix Client)
+  # ======================
+  element-web:
+    image: vectorim/element-web:latest
+    container_name: mosaic-element-web
+    restart: unless-stopped
+    ports:
+      - "${ELEMENT_PORT:-8501}:80"
+    volumes:
+      - ./matrix/element/config.json:/app/config.json:ro
+    depends_on:
+      synapse:
+        condition: service_healthy
+    healthcheck:
+      test: ["CMD-SHELL", "wget --no-verbose --tries=1 --spider http://localhost:80 || exit 1"]
+      interval: 30s
+      timeout: 5s
+      retries: 3
+      start_period: 10s
+    networks:
+      - mosaic-network
+    labels:
+      com.mosaic.service: "matrix-element"
+      com.mosaic.description: "Element Web client (dev)"
+
+volumes:
+  synapse_data:
+    name: mosaic-synapse-data
+  synapse_signing_key:
+    name: mosaic-synapse-signing-key