feat(coordinator): worker registry, API contract, logging & DB retry

Align the coordinator with the master PLAN.md (CTX-00, CTX-04) and harden
process startup.

- CTX-00: freeze docs/api-contract.md as the v1 source of truth for the
  Go coordinator and Python worker.
- CTX-04: worker registry — workers table (migration 0002), domain.Worker,
  RegisterWorker use case, WorkerRepository, and POST /workers/register.
- Contract alignment: claim uses `capabilities` (was `workloads`),
  COORDINATOR_TOKEN env (WORKER_AUTH_TOKEN kept as fallback), and
  GET /health now reports database readiness (503 when the DB is down).
- Logging: logs are teed to stdout and an optional rotated file (LOG_FILE)
  via lumberjack, so they survive a container rebuild.
- Startup resilience: the initial DB connection is retried with backoff,
  so the coordinator waits for Postgres to boot instead of crash-looping.

coordinator/internal/usecase/dto.go

@@ -21,6 +21,11 @@ type ChunkInput struct {
 	MaxAttempts int
 }
 
+type RegisterWorkerInput struct {
+	Name         string
+	Capabilities []string
+}
+
 type ClaimTaskInput struct {
 	WorkerID  string
 	Workloads []string

coordinator/internal/usecase/ports.go

@@ -62,6 +62,12 @@ type JobRepository interface {
 	UpdateStatus(ctx context.Context, id uuid.UUID, status domain.JobStatus, completedAt *time.Time) error
 }
 
+// WorkerRepository persists the worker registry.
+type WorkerRepository interface {
+	Insert(ctx context.Context, w *domain.Worker) error
+	Get(ctx context.Context, id uuid.UUID) (*domain.Worker, error)
+}
+
 // TxManager runs a function inside one database transaction. The transaction
 // travels in the context, so repositories pick it up without this port ever
 // mentioning pgx.

coordinator/internal/usecase/worker.go

@@ -0,0 +1,28 @@
+package usecase
+
+import (
+	"context"
+
+	"github.com/emil28092005/SciMesh/coordinator/internal/domain"
+)
+
+// RegisterWorker records a worker in the registry and hands back its identity.
+type RegisterWorker struct {
+	workers WorkerRepository
+	clk     Clock
+}
+
+func NewRegisterWorker(workers WorkerRepository, clk Clock) *RegisterWorker {
+	return &RegisterWorker{workers: workers, clk: clk}
+}
+
+func (uc *RegisterWorker) Execute(ctx context.Context, in RegisterWorkerInput) (*domain.Worker, error) {
+	w, err := domain.NewWorker(in.Name, in.Capabilities, uc.clk.Now())
+	if err != nil {
+		return nil, err
+	}
+	if err := uc.workers.Insert(ctx, w); err != nil {
+		return nil, err
+	}
+	return w, nil
+}