fix(cache-warmer): tighten warming heuristics to reduce net negative spend

[BYK]

Problem

Cache warming was net negative: $11.18 spent on warmup pings but only $5.52 saved from cache hits (net loss: $5.66). The system was too eager — warming sessions that don't return, warming too many cycles during tool calls, and accepting trivially low return probabilities.

Root Causes (from live dashboard data)

Issue	Example	Impact
Initial threshold too low (4-9%)	Sessions with 2.1% P(returns) got warmed	Nearly every session gets first warmup
Tool-call warming uncapped	31 warmups on one session (52% hit rate)	Excessive spend during tool chains
Small sessions warmed too eagerly	5-turn session got 6 warmups, 0 hits	Poor ROI on short sessions
ROI check kicks in too late	2/11 hit rate — stopped at warmup #11	Wasted $0.50+ before cutoff

Fixes

1. Initial commitment threshold floor (HIGHEST IMPACT)

P(returns) must now exceed 30% (was 4-9% break-even ratio). Prevents cascading continuation chains on marginal sessions.

2. Tool-call warming cap

• New TOOL_CALL_MAX_CYCLES = 2 (was unlimited up to break-even)
• MAX_TOOL_CALL_WARMING_MS: 30min -> 10min
• Most tool calls complete in <5 min; 2 cycles covers 10 min operations

3. Higher minimum turn count

MIN_TURNS_FOR_WARMING: 3 -> 5 (requires 10 messages). Ensures sufficient survival model data.

4. Minimum context size gate

New MIN_INPUT_TOKENS_FOR_WARMING = 50,000. Below this, absolute savings per hit are too small to justify risk. Force-keep path is exempt.

5. Tighter session-level ROI check

• MIN_WARMUPS_FOR_ROI_CHECK: 10 -> 5 (kicks in sooner)
• MIN_SESSION_HIT_RATE: 20% -> 25% (stricter bar)

Files Changed

• packages/gateway/src/cache-warmer.ts — 5 constant changes, 3 new gates in shouldWarm(), dashboard reason strings
• packages/gateway/src/idle.ts — context size early-exit + import
• packages/gateway/test/cache-warmer.test.ts — 3 existing tests updated, 10 new tests
• packages/gateway/test/helpers/idle-worker.ts — mock updated for new exports

Expected Impact

Estimated ~37-49 fewer wasted warmups per cycle, saving $3.70-4.90 against the $5.66 net loss. Should flip warming to net positive as only profitable sessions get warmed.

What we're NOT changing

• pSessionFinished() base rate — the 30% threshold floor addresses the same problem more directly
• Warming overall — historical data shows $369 in warming savings from 775 hits across 964 sessions; the system is profitable at scale with better heuristics