feat(memory): hybrid vector search via embedder

- repository: optional pgvector embedding column (migration with
  CREATE EXTENSION guarded by main), SearchByVector cosine search,
  vector-enabled insert path
- service: SetEmbedder — memories are embedded on Add; Search merges
  vector hits with n-gram keyword matches (vector first, deduped)
- embedder was already implemented (OpenAI-compatible API + n-gram FNV
  fallback) but never instantiated; main now wires it from
  EMBEDDING_API_URL/EMBEDDING_MODEL config

backend/services/memory/repository.go

@@ -12,7 +12,8 @@ import (
 
 // Repository handles database operations for memories.
 type Repository struct {
-	db *sql.DB
+	db            *sql.DB
+	vectorEnabled bool // true when the pgvector embedding column exists
 }
 
 // NewRepository creates a new memory repository.
@@ -20,6 +21,36 @@ func NewRepository(db *sql.DB) *Repository {
 	return &Repository{db: db}
 }
 
+// EnableVector marks the embedding column as available (after migration).
+func (r *Repository) EnableVector() { r.vectorEnabled = true }
+
+// VectorEnabled reports whether vector search is usable.
+func (r *Repository) VectorEnabled() bool { return r.vectorEnabled }
+
+// VectorMigrationSQL returns the statements to enable pgvector and add the
+// embedding column with the given dimension.
+func VectorMigrationSQL(dims int) string {
+	return fmt.Sprintf(`
+	CREATE EXTENSION IF NOT EXISTS vector;
+	ALTER TABLE memories ADD COLUMN IF NOT EXISTS embedding vector(%d);
+	CREATE INDEX IF NOT EXISTS idx_memories_embedding ON memories USING hnsw (embedding vector_cosine_ops);
+	`, dims)
+}
+
+// vectorString formats an embedding for pgvector input.
+func vectorString(vec []float64) string {
+	var sb strings.Builder
+	sb.WriteByte('[')
+	for i, v := range vec {
+		if i > 0 {
+			sb.WriteByte(',')
+		}
+		fmt.Fprintf(&sb, "%g", v)
+	}
+	sb.WriteByte(']')
+	return sb.String()
+}
+
 // Create inserts a new memory and returns its ID.
 func (r *Repository) Create(ctx context.Context, m *Memory) (string, error) {
 	query := `
@@ -41,6 +72,24 @@ func (r *Repository) Create(ctx context.Context, m *Memory) (string, error) {
 	}
 
 	var id string
+	if r.vectorEnabled && len(m.Embedding) > 0 {
+		query = `
+			INSERT INTO memories (user_id, source_id, content, summary, keywords, category, priority, decay, access_count, last_accessed_at, created_at, updated_at, embedding)
+			VALUES ($1, $2, $3, $4, $5, $6, $7, $8, $9, $10, $11, $12, $13::vector)
+			RETURNING id
+		`
+		err := r.db.QueryRowContext(ctx, query,
+			m.UserID, m.SourceID, m.Content, m.Summary,
+			pq.Array(m.Keywords), m.Category, m.Priority, m.Decay,
+			m.AccessCnt, m.LastAcc, m.CreatedAt, m.UpdatedAt, vectorString(m.Embedding),
+		).Scan(&id)
+		if err != nil {
+			return "", fmt.Errorf("insert memory: %w", err)
+		}
+		m.ID = id
+		return id, nil
+	}
+
 	err := r.db.QueryRowContext(ctx, query,
 		m.UserID, m.SourceID, m.Content, m.Summary,
 		pq.Array(m.Keywords), m.Category, m.Priority, m.Decay,
@@ -127,6 +176,22 @@ func (r *Repository) SearchByNGrams(ctx context.Context, userID string, nGrams [
 	return r.queryMemories(ctx, query, userID, DecayThreshold, pq.Array(nGrams), limit)
 }
 
+// SearchByVector returns the nearest memories by cosine distance.
+func (r *Repository) SearchByVector(ctx context.Context, userID string, vec []float64, limit int) ([]Memory, error) {
+	if !r.vectorEnabled || len(vec) == 0 {
+		return nil, nil
+	}
+	if limit <= 0 {
+		limit = 10
+	}
+	query := `
+		SELECT id, user_id, source_id, content, summary, keywords, category, priority, decay, access_count, last_accessed_at, created_at, updated_at
+		FROM memories WHERE user_id = $1 AND decay > $2 AND embedding IS NOT NULL
+		ORDER BY embedding <=> $3::vector LIMIT $4
+	`
+	return r.queryMemories(ctx, query, userID, DecayThreshold, vectorString(vec), limit)
+}
+
 // Update updates an existing memory.
 func (r *Repository) Update(ctx context.Context, m *Memory) error {
 	m.UpdatedAt = time.Now()