feat(auth): publish the resolved Coach reasoning on the account payload

GET /api/auth/me gains coachReasoning: the level a turn runs at, the
person's own pick, the highest level the settings may offer, whether
reasoning can run at all, whether the lowest option is a real off, and
the layer that decided. Resolved on the server so the web settings and
the native client read the same answer.

The web reads it through useCoachReasoning. The Coach preferences route
names the stored level on its wide event, and a unit test round-trips
the level through the account-settings backup section.

docs/api/openapi.yaml

@@ -74400,6 +74400,8 @@ components:
           $ref: "#/components/schemas/ModuleAccessMap"
         ai:
           $ref: "#/components/schemas/AiAccountBlock"
+        coachReasoning:
+          $ref: "#/components/schemas/CoachReasoningState"
         recordSession:
           anyOf:
             - $ref: "#/components/schemas/RecordSessionState"
@@ -74426,6 +74428,7 @@ components:
         - accountAccess
         - moduleAccess
         - ai
+        - coachReasoning
         - recordSession
         - notificationPrefs
         - onboarding
@@ -74440,7 +74443,9 @@ components:
         carries none — the two are the same account and nothing is masked. `moduleAccess` says WHY each module is or is
         not there — the record's own switch, the edge of the grant, or the operator's instance-wide one — beside the
         `modules` booleans it never contradicts. `ai` (v1.39) is the resolved AI capability map for the same record,
-        with the same masking; it replaces reading `GET /api/feature-flags`."
+        with the same masking; it replaces reading `GET /api/feature-flags`. `coachReasoning` (v1.41) is the Coach's
+        resolved thinking depth: the caller's own preference against the operator's reasoning controls and the record's
+        provider."
     AccountAccess:
       type: object
       properties:
@@ -74933,6 +74938,67 @@ components:
         - responseTimeoutMs
       additionalProperties: false
       description: The account-level companion to the capability map.
+    CoachReasoningState:
+      type: object
+      properties:
+        level:
+          type: string
+          enum:
+            - off
+            - low
+            - medium
+            - high
+          description: "The level a Coach turn runs at right now, every cap applied: the person's choice, lowered to the
+            operator's highest level, to `medium` when the operator pays for the provider, and `off` when the operator
+            switched reasoning off or the provider cannot reason."
+        preference:
+          type: string
+          enum:
+            - off
+            - low
+            - medium
+            - high
+          description: What the person picked (`reasoning` on `PUT /api/auth/me/coach-prefs`), `medium` when they never chose.
+            Bind the settings field to this, not to `level`.
+        maxLevel:
+          type: string
+          enum:
+            - off
+            - low
+            - medium
+            - high
+          description: The highest level the settings may offer. Options above it are shown disabled as limited by the operator.
+            `off` when the operator switched reasoning off.
+        available:
+          type: boolean
+          description: False when the operator switched reasoning off or the configured provider cannot reason; the settings field
+            is then locked.
+        offIsReal:
+          type: boolean
+          description: False when the provider cannot switch reasoning off (it thinks at its lowest level instead); label the
+            lowest option "Minimal" rather than "Off".
+        source:
+          type: string
+          enum:
+            - user
+            - job
+            - admin_off
+            - admin_cap
+            - cost_cap
+            - unsupported
+          description: "Which layer decided `level`: the person's own choice (`user`), the operator's switch (`admin_off`) or cap
+            (`admin_cap`), the operator-funded ceiling (`cost_cap`), or a provider that cannot reason (`unsupported`).
+            `job` never appears here."
+      required:
+        - level
+        - preference
+        - maxLevel
+        - available
+        - offIsReal
+        - source
+      additionalProperties: false
+      description: The Coach's thinking depth for the signed-in person, resolved on the server from their preference, the
+        operator's reasoning switch and cap, who pays for the provider, and what the provider can do.
     RecordSessionState:
       type: object
       properties: