phsl / new-api

Deferred follow-ups from the relaykit-tools review cycle, verified by
live end-to-end billing tests:

- billing: normalize Gemini modality keys consistently between stream
  merge and settlement (case/whitespace variants no longer drop
  independent audio/image pricing) and sum duplicate modality entries
  on both paths
- billing: sync legacy flat Claude cache-creation fields from the
  CacheCreation sub-object (including zeroing) and fall back to flat
  fields only when the snapshot never carried a sub-object, closing a
  stale 1h-cache overcharge path in cascaded deployments
- relay: move Chat-to-Claude and Chat-to-Gemini stream conversion state
  from gin.Context onto RelayInfo and reset it with SendResponseCount in
  InitChannelMeta, so channel retries start clean while per-request
  state (stream error collection, conversion diagnostics, channel
  chain, billing accumulators) survives
- relay: Claude channel now serves Gemini-format clients (request via
  registry conversion, response and stream composed through the Chat
  pivot), removing the last unimplemented conversion direction
- relaykit: recognize legacy pseudo tool names (googleSearch,
  codeExecution, urlContext) in the toolconv decode stage and drop the
  string-matching bypass in the Chat-to-Gemini converter; native Gemini
  tool output is restored and non-Gemini targets follow standard loss
  diagnostics
- relaykit: attach upstream Gemini usage (with billing_usage sidecar)
  to intermediate stream chunks so converted Claude streams report
  upstream truth from message_start, and preserve the sidecar through
  Claude stream usage merges; billing settlement unchanged
- billing: clamp negative Total-Prompt completion derivation, OR the
  Estimated flag across cross-dialect snapshot replacement, and fill
  canonical OpenAI prompt details via field-wise merge