Slice B6b: Add Gate 4 automated startup script

New file: scripts/gate-4-startup.ps1
- Automated host startup sequence (DEBUG mode)
- Prerequisite validation (PostgreSQL connectivity, .NET SDK)
- Optional DbUp migration execution
- Environment variable setup (KRX_OPENAPI stub, KARTSELL_POSTGRES)
- Usage: .\scripts\gate-4-startup.ps1 [-SkipDbUp] [-Environment Debug|Release]

AGENTS.md v16.0 automation: DRY principle (eliminates manual terminal steps)
Gate 4 readiness: Complete (build ✅, tests ✅, script ✅)

Co-Authored-By: Claude Haiku 4.5 <noreply@anthropic.com>

scripts/gate-4-startup.ps1

@@ -0,0 +1,67 @@
+# Gate 4: Complete Host Startup Sequence
+# AGENTS.md v16.0 Strategic Automation
+# Prerequisites: SSH tunnel open, DbUp migrations applied
+
+param(
+    [switch]$SkipDbUp = $false,
+    [string]$Environment = "Debug"
+)
+
+Write-Host "=== Gate 4: Host Startup Sequence ===" -ForegroundColor Green
+Write-Host "Environment: $Environment (Debug = DevelopmentHeaderAuthenticationHandler)" -ForegroundColor Cyan
+Write-Host ""
+
+# Step 1: Verify Prerequisites
+Write-Host "[1/3] Verifying prerequisites..." -ForegroundColor Yellow
+
+# Check PostgreSQL connection
+try {
+    $testConn = New-Object System.Net.Sockets.TcpClient
+    $testConn.Connect("localhost", 5432)
+    $testConn.Close()
+    Write-Host "  ✅ PostgreSQL available (localhost:5432)" -ForegroundColor Green
+}
+catch {
+    Write-Host "  ❌ PostgreSQL not accessible. Start SSH tunnel:" -ForegroundColor Red
+    Write-Host "     ssh -L 5432:127.0.0.1:5432 kjh2064@178.104.200.7" -ForegroundColor Yellow
+    exit 1
+}
+
+# Check .NET SDK
+$dotnetVersion = dotnet --version
+Write-Host "  ✅ .NET SDK available ($dotnetVersion)" -ForegroundColor Green
+
+# Step 2: Run Migrations (if not skipped)
+if (-not $SkipDbUp) {
+    Write-Host "[2/3] Running database migrations..." -ForegroundColor Yellow
+    $env:KARTSELL_POSTGRES = "Host=localhost;Port=5432;Database=kartsell;Username=kartsell;Password=kartsell"
+
+    try {
+        dotnet run --project src/KArtSell.DbMigrator -c Release --no-build
+        Write-Host "  ✅ Migrations applied successfully" -ForegroundColor Green
+    }
+    catch {
+        Write-Host "  ⚠️  Migration warning: $_" -ForegroundColor Yellow
+    }
+}
+
+# Step 3: Start Host
+Write-Host "[3/3] Starting Host (DEBUG mode)..." -ForegroundColor Yellow
+Write-Host ""
+
+# Set required environment variables
+$env:KARTSELL_POSTGRES = "Host=localhost;Port=5432;Database=kartsell;Username=kartsell;Password=kartsell"
+$env:KRX_OPENAPI = "stub-key-for-testing"
+
+# Display startup info
+Write-Host "Configuration: $Environment" -ForegroundColor Cyan
+Write-Host "Expected output: 'Now listening on: http://127.0.0.1:5002'" -ForegroundColor Cyan
+Write-Host ""
+Write-Host "Press Ctrl+C to stop." -ForegroundColor Gray
+Write-Host ""
+
+# Launch Host
+dotnet run --project src/KArtSell.Host --configuration $Environment --no-build
+
+Write-Host ""
+Write-Host "Host stopped." -ForegroundColor Yellow