Add self-service sign-up and invite links

First half of #7. Until now the only way to get an account was for
somebody who already had one to create it, and the login page told people
to "ask an admin" -- workable for one operator, impossible for a team.

Two modes, chosen by an administrator in the settings table: invite_only,
which is the default, and open. A third domain-restricted mode was
considered and dropped, because with no email in this server there is
nothing to verify an address against and it would only check the domain
of a string somebody typed.

The default is the closed door. An install that gets a public hostname
before anybody has thought about sign-up should not be collecting
accounts from the internet, and the failure mode of a typo in the setting
is invite_only rather than open.

An invite is a link, not an email. Adding SMTP to send one message would
be a subsystem to run, secure and monitor; the person inviting sends the
link however they already talk to the person they are inviting. A link
carries the team and the role, because an account in no team sees an
empty queue and can be paged by nobody -- that is not a state to invite
somebody into. Links are single-use by default, expire after seven days,
and can be revoked before that: a link that works forever is a credential
nobody remembers issuing, sitting in a chat log.

The uses counter is incremented inside the sign-up transaction and
guarded by `uses < max_uses`, so two people redeeming the last use at
once cannot both get in.

GET /api/signup reports the mode and whether a link is usable, so the
form can say "this link has expired" before somebody picks a password
rather than after. It gives one answer for expired, revoked, used up and
never existed: telling a stranger which it was tells them something about
links they do not hold.

Sign-up signs you in. The alternative is a form that says "now go and log
in", which is the same credential typed twice. login and signup now share
startSession rather than each minting a cookie.

Rate-limited per address on its own limiter, not login's: a burst of
sign-ups must not lock somebody out of logging in.

The settings table grew a second shape for this. It held only durations;
signup_mode is a word from a fixed list, so the admin endpoint now
validates everything before writing anything -- a request that sets two
settings and gets one wrong changes neither.

Still to come in #7: the sign-up and invite-redemption pages, the
first-run checklist, and the in-app integration instructions. The schema
carries onboarding_dismissed_at for the checklist already.

Claude-Session: https://claude.ai/code/session_01RHPj4ggeFdEjKKfm4SHbD7

internal/api/router.go

@@ -15,6 +15,12 @@ import (
 // notify disables notifications. Dead man's switches are per team and read from
 // the database, so nothing about them is wired in here.
 func NewRouter(db *sql.DB, notify NotifyConfig, cfg config.Config) http.Handler {
+	// One limiter each, both process-wide for the life of the router: login
+	// counts failed passwords, sign-up counts account creation, and mixing the
+	// two would let a burst of sign-ups lock somebody out of logging in.
+	loginLimit := newLoginLimiter()
+	signupLimiter := newLoginLimiter()
+
 	r := chi.NewRouter()
 	r.Use(middleware.Logger)
 	r.Use(middleware.Recoverer)
@@ -39,9 +45,16 @@ func NewRouter(db *sql.DB, notify NotifyConfig, cfg config.Config) http.Handler
 	// JSON 404 every unknown /api path gets.
 	r.Post("/api/integrations/{key}/alertmanager", handleIntegrationWebhook(db, notify))
 
+	// Signing up. Both are unauthenticated by necessity: the caller has no
+	// account yet. The info endpoint says whether the door is open and whether
+	// an invite link is good, so the form can say so before somebody picks a
+	// password.
+	r.Get("/api/signup", handleSignupInfo(db))
+	r.Post("/api/signup", handleSignup(db, signupLimiter, notify.PublicURL))
+
 	// Signing in to the web UI. Login trades a password for a session cookie,
 	// which AuthMiddleware accepts in place of an API key.
-	r.Post("/api/login", handleLogin(db, newLoginLimiter(), notify.PublicURL))
+	r.Post("/api/login", handleLogin(db, loginLimit, notify.PublicURL))
 	r.Post("/api/logout", handleLogout(db, notify.PublicURL))
 
 	// All other /api routes require a valid API key.
@@ -109,6 +122,11 @@ func NewRouter(db *sql.DB, notify NotifyConfig, cfg config.Config) http.Handler
 		r.Post("/api/teams/{teamID}/members", handleAddTeamMember(db))
 		r.Delete("/api/teams/{teamID}/members/{userID}", handleRemoveTeamMember(db))
 
+		// Invite links into this team.
+		r.Get("/api/teams/{teamID}/invites", handleListInvites(db))
+		r.Post("/api/teams/{teamID}/invites", handleCreateInvite(db, notify.PublicURL))
+		r.Delete("/api/teams/{teamID}/invites/{inviteID}", handleRevokeInvite(db))
+
 		// A team's escalation ladder: who is paged when nobody answers.
 		r.Get("/api/teams/{teamID}/escalation", handleGetEscalation(db))
 		r.Put("/api/teams/{teamID}/escalation", handleSetEscalation(db))