Add a reasoning-off setting (-1 reasoning budget)

Models like DeepSeek V4 Flash reason by default, and the reasoning budget
setting could only ever add thinking tokens - there was no value that turned
thinking off. A negative budget now sends `reasoning: {effort: "none"}`.

Uses effort:none rather than exclude:true deliberately - exclude still thinks
and still bills, it only hides the trace.

Zero keeps its old meaning (send no `reasoning` field at all) so endpoints that
reject unknown fields, like the default Ollama one, are unaffected. Reusing the
existing int column this way avoids a migration.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01UeQVy5bEjLhfgWNc27Efet

backend/tests/test_reasoning_param.py

@@ -0,0 +1,45 @@
+"""What the provider puts in the `reasoning` request field for each budget
+setting: a positive budget asks for thinking, 0 stays silent, -1 turns it off.
+
+    python -m pytest tests/test_reasoning_param.py -v
+"""
+from app.providers.openai_compatible import OpenAICompatibleProvider
+
+
+def _body(reasoning_max_tokens, api_mode="chat", max_tokens=1000):
+    provider = OpenAICompatibleProvider(
+        "https://openrouter.ai/api/v1", "k", "deepseek/deepseek-v4-flash-0731",
+        api_mode, reasoning_max_tokens,
+    )
+    body = {"max_tokens": max_tokens}
+    provider._apply_reasoning_budget(body)
+    return body
+
+
+def test_zero_sends_nothing():
+    """Ollama and friends reject unknown fields — 0 must stay silent."""
+    assert "reasoning" not in _body(0)
+
+
+def test_positive_budget_adds_thinking_tokens():
+    body = _body(500)
+    assert body["reasoning"] == {"max_tokens": 500}
+    # the story output keeps its own full budget on top of the thinking budget
+    assert body["max_tokens"] == 1500
+
+
+def test_negative_turns_reasoning_off():
+    body = _body(-1)
+    assert body["reasoning"] == {"effort": "none"}
+    # "off" must not inflate the output budget
+    assert body["max_tokens"] == 1000
+
+
+def test_off_is_not_merely_excluded():
+    """`exclude: true` still thinks and still bills; we want it actually off."""
+    assert _body(-1)["reasoning"].get("exclude") is None
+
+
+def test_completion_mode_never_sends_reasoning():
+    for budget in (-1, 0, 500):
+        assert "reasoning" not in _body(budget, api_mode="completion")