Add DAB ANTCAP calibration and fix BT1035 boot banner timing

Extend the FM-only ANTCAP antenna-varactor override to DAB, mirroring the
existing mechanism end to end (driver, tuner, service, EEPROM storage,
HTTP API). Live sweep on real hardware found no ANTCAP value beating
auto-tune on the ensembles tested, so DAB stays on auto-tune by default.

Also fix BT1035 boot: the module's real boot banner doesn't appear until
~18-24s after RESET# releases, not the 3.5s previously waited; add a
2-attempt retry and a baud-rate probe fallback for diagnostics.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

Software/docs/manual/ch-api.tex

@@ -6,7 +6,7 @@ implemented in \texttt{SetupWebServer}. Request bodies are parsed into
 domain types in the pure core (\texttt{components/core}) before any
 persistence or driver call. Exact C++ signatures live in the generated
 Doxygen output under \texttt{docs/api/}; this chapter documents the
-wire protocol and behaviour as shipped in firmware~0.8.5.
+wire protocol and behaviour as shipped in firmware~0.9.0.
 
 \section{Transport and reachability}
 
@@ -36,7 +36,7 @@ Returns a health-check DTO serialised by
 
 \begin{drnote}[Response schema]
 \begin{drcode}[JSON]
-{"status":"ok","fw":"0.8.5","serialNumber":"0004A3123456",
+{"status":"ok","fw":"0.9.0","serialNumber":"0004A3123456",
  "chips":{"si4684":true,"adau1701":true,"bt1035":true}}
 \end{drcode}
 \begin{itemize}
@@ -172,7 +172,13 @@ or
 {"band":"fm","frequency_khz":101500}
 \end{drcode}
 DAB \texttt{freq\_index} is 0--37; FM \texttt{frequency\_khz} is
-64\,000--108\,000.
+64\,000--108\,000. FM accepts an optional \texttt{antcap} (0--128):
+overrides the front-end antenna varactor for this one tune, e.g. while
+running a calibration sweep. Omit it (the normal case) to use the board's
+saved calibration --- see
+\texttt{POST /api/tuner/calibrate-antenna}
+(Section~\ref{sec:api-tuner-calibrate-antenna}) --- or hardware auto-tune
+if never calibrated. Ignored for \texttt{"dab"}.
 \end{drnote}
 
 On success returns the updated status JSON (same shape as
@@ -267,6 +273,33 @@ per-station poll budget ran out.
 HTTP status: \textbf{200 OK}; \textbf{409} on a hardware/driver failure
 mid-sweep; \textbf{503} when the tuner service is unavailable.
 
+\apiendpoint{POST}{/api/tuner/calibrate-antenna}
+\label{sec:api-tuner-calibrate-antenna}
+
+Commits an FM ANTCAP value (AN851 Appendix A front-end calibration) as the
+board's permanent default, found by sweeping \texttt{antcap} on
+\texttt{POST /api/tuner/tune} (Section~\ref{sec:api-tuner-tune}) across
+0--128 and comparing \texttt{rssi\_dbuv}/\texttt{snr\_db}. Persists to the
+24AA025E48 EEPROM's user-writable region (separate from the factory-locked
+EUI-48) and takes effect immediately on the live tuner --- no reboot
+required, though it also survives one since \texttt{HardwareBootstrap::boot()}
+reloads it at every boot. Once saved, every ordinary FM tune (seek, scan,
+station recall, live UI) uses this value automatically instead of the
+chip's own front-end auto-tune, unless a request explicitly overrides
+\texttt{antcap} for that one call.
+
+\begin{drnote}[Request schema]
+\begin{drcode}[JSON]
+{"antcap":102}
+\end{drcode}
+\texttt{antcap} is required, 0--128.
+\end{drnote}
+
+Success response: \texttt{\{"status":"saved","antcap":102\}}. HTTP status:
+\textbf{200 OK}; \textbf{400} for invalid/missing \texttt{antcap};
+\textbf{500} on an EEPROM write failure; \textbf{503} when the tuner
+service is unavailable.
+
 \apiendpoint{GET}{/api/audio/profile}
 \label{sec:api-audio-profile-get}