refactor(faces): keep the generated contributions out of the Host aggregate

The Gateway and the carrier each compiled both halves under one tsconfig, so
the Host aggregate built their browser faces — including the face that owns
`ctx.remote`, the most likely future consumer of a generated `/remote`
contribution. Both packages now expose a host and a client face, and each
aggregate references only its own; three modules the halves share appear in
both file lists, as api/remotes already does.

The two apps/web specs in the Host aggregate restate the conversation engine's
Context key format instead of importing the Client runtime for it. A drift
makes the key miss its rendered node, so the assertion fails loudly.

The Host aggregate now reaches one Client project, the carrier's host face,
which the Gateway's own dispatch face needs; no generated contribution is
reachable from it.

tsconfig.client.json

@@ -19,10 +19,7 @@
     "packages/client/*/src/css-modules.d.ts",
     "packages/client/*/tests/**/*.ts",
     "packages/client/*/tests/**/*.tsx",
-    "packages/host/directory-picker-browse/tests/**/*.ts",
-    "packages/host/directory-picker-browse/tests/**/*.tsx",
-    "packages/host/directory-picker-native/tests/**/*.ts",
-    "packages/host/directory-picker-native/tests/**/*.tsx",
+    "packages/api/gateway/tests/client.spec.ts",
     "packages/client/tsdown.client.ts",
     "scripts/client-bundle-css.spec.ts",
     "scripts/client-bundle-purity.spec.ts"
@@ -32,11 +29,6 @@
     // smoke policy). webserver has zero workspace deps and no cordis merge,
     // so it cannot drag host-side Context augmentation into this program.
     { "path": "./packages/host/webserver" },
-    // Dual-face host leaf: the node half is the native picking backend, the
-    // browser half registers the picking flow into ui-workspace's slot —
-    // client-side Context merges keep it out of the host program.
-    { "path": "./packages/host/directory-picker-native" },
-    { "path": "./packages/host/directory-picker-browse" },
     // Compaction seam: the client-runtime pin test value-imports the canonical
     // checkpoint const from the cordis-free dsh-compact/checkpoint leaf and
     // deliberately never loads the dsh-compact package root or the host-side
@@ -51,9 +43,12 @@
     { "path": "./packages/client/web-react" },
     { "path": "./packages/client/modules" },
     { "path": "./packages/client/hmr" },
-    { "path": "./packages/client/connection" },
+    { "path": "./packages/client/connection/tsconfig.client.json" },
+    // The carrier's node-half spec rides this aggregate's package test glob, so
+    // its Host face is referenced here too — the mirror of the webserver leaf.
+    { "path": "./packages/client/connection/tsconfig.host.json" },
     { "path": "./packages/typert/registry" },
-    { "path": "./packages/api/gateway" },
+    { "path": "./packages/api/gateway/tsconfig.client.json" },
     { "path": "./packages/api/remotes/tsconfig.client.json" },
     { "path": "./packages/client/runtime" },
     { "path": "./packages/client/test-runtime" },
@@ -69,6 +64,8 @@
     { "path": "./packages/client/ui-skill" },
     { "path": "./packages/client/ui-subagent" },
     { "path": "./packages/client/ui-task" },
+    { "path": "./packages/client/ui-directory-picker" },
+    { "path": "./packages/client/ui-directory-picker-native" },
     { "path": "./packages/client/ui-goal" },
     { "path": "./packages/client/ui-model" },
     { "path": "./packages/client/ui-agent-preset" },