fix(adaptive_router): merge persisted deltas onto cold-start prior on reload

AdaptiveRouter.load_state_from_db overwrote each cold-start Beta prior with the raw
persisted row, but AdaptiveRouterUpdateQueue.flush_state_to_db persists accumulated
DELTAS (Prisma increment, for multi-pod safety). A satisfaction-only cell persists
beta=0 (a negative-only cell persists alpha=0); on the next proxy restart
thompson_sample -> random.betavariate(alpha, beta) with a 0 argument raises
"ValueError: gammavariate: alpha and beta must be > 0.0", returning HTTP 500 for every
request whose request_type has such a cell (pick_best samples every pool model's cell).

Merge the persisted deltas onto the cold-start prior instead of overwriting it: keeps the
Beta valid (prior.beta >= 0.5) and preserves the tier-seeded prior across restarts.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

litellm/router_strategy/adaptive_router/adaptive_router.py

@@ -109,7 +109,7 @@ class AdaptiveRouter:
                 self._cells[(rt, model)] = initial_cell(prefs, rt)
 
     async def load_state_from_db(self, prisma_client: Any) -> None:
-        """Override cold-start cells with persisted state. Called once at startup."""
+        """Merge persisted deltas onto the cold-start prior. Called once at startup."""
         if prisma_client is None:
             return
         try:
@@ -125,8 +125,16 @@ class AdaptiveRouter:
                     continue
                 if row.model_name not in self.config.available_models:
                     continue
+                # Persisted alpha/beta are accumulated DELTAS — the flusher writes
+                # increments, not absolute posteriors (for multi-pod safety). Merge
+                # them onto the cold-start prior so the tier bias is preserved AND the
+                # Beta stays valid (prior.beta >= 0.5, so a satisfaction-only delta of
+                # beta=0 can never reload as Beta(alpha, 0) and crash thompson_sample).
+                prefs = self.model_to_prefs.get(row.model_name) or _default_prefs()
+                prior = initial_cell(prefs, rt)
                 self._cells[(rt, row.model_name)] = BanditCell(
-                    alpha=row.alpha, beta=row.beta
+                    alpha=prior.alpha + row.alpha,
+                    beta=prior.beta + row.beta,
                 )
                 loaded += 1
             verbose_router_logger.info(