feat(beacons): plot-path link seeds TX chain to match beacon EIRP

The "Plot path to beacon" action now includes `tx_power_dbm` and
`src_gain_dbi` in the /path query string so the path calculator's TX
side emits the same EIRP as the beacon out of the box. The beacon
schema stores `power_mw` as EIRP already (per Beacons.RangeEstimate),
so we take 10·log10(power_mw), split against a default 30 dBi dish,
and clamp the TX power to a 0 dBm floor so very-low-EIRP beacons
produce a sane gain/power split instead of -10 dBm outputs.

Examples:
  10 W EIRP (40 dBm)  → 30 dBi dish + 10 dBm TX
  100 W EIRP (50 dBm) → 30 dBi dish + 20 dBm TX
  100 mW EIRP (20 dBm) → 20 dBi + 0 dBm (clamped)

lib/microwaveprop_web/live/beacon_live/show.ex

@@ -285,11 +285,41 @@ defmodule MicrowavepropWeb.BeaconLive.Show do
       "dst_height_ft" => to_string(beacon.height_ft)
     }
 
-    params = maybe_put(base, "band", BandResolver.resolve_as_string(beacon.frequency_mhz))
+    {tx_power_dbm, src_gain_dbi} = baseline_tx_for_eirp(beacon.power_mw)
+
+    params =
+      base
+      |> maybe_put("band", BandResolver.resolve_as_string(beacon.frequency_mhz))
+      |> maybe_put("tx_power_dbm", tx_power_dbm)
+      |> maybe_put("src_gain_dbi", src_gain_dbi)
 
     "/path?" <> URI.encode_query(params)
   end
 
+  # Splits the beacon's EIRP (the beacon schema's `power_mw` field is
+  # already an EIRP value, per Beacons.RangeEstimate) into a
+  # tx_power_dbm + src_gain_dbi pair so the path calculator's TX chain
+  # emits the same EIRP as the beacon. We default the antenna gain to
+  # 30 dBi (typical 2-3 ft microwave dish) and derive the TX power from
+  # there, clamping to a 0 dBm / 1 mW minimum so very-low-EIRP beacons
+  # produce sane baseline values instead of -10 dBm absurdities.
+  defp baseline_tx_for_eirp(nil), do: {nil, nil}
+  defp baseline_tx_for_eirp(power_mw) when power_mw <= 0, do: {nil, nil}
+
+  defp baseline_tx_for_eirp(power_mw) do
+    eirp_dbm = 10.0 * :math.log10(power_mw)
+    default_gain = 30.0
+    min_tx_dbm = 0.0
+
+    tx_dbm = max(min_tx_dbm, eirp_dbm - default_gain)
+    gain_dbi = eirp_dbm - tx_dbm
+
+    {format_dbm(tx_dbm), format_dbi(gain_dbi)}
+  end
+
+  defp format_dbm(v), do: v |> Float.round(1) |> to_string()
+  defp format_dbi(v), do: v |> Float.round(1) |> to_string()
+
   defp precise_grid(%Beacon{lat: lat, lon: lon}) when is_number(lat) and is_number(lon) do
     Maidenhead.from_latlon(lat, lon, 8)
   end

test/microwaveprop_web/live/beacon_live_test.exs

@@ -92,6 +92,25 @@ defmodule MicrowavepropWeb.BeaconLiveTest do
       {:ok, _view, html} = live(conn, ~p"/beacons/#{beacon}")
       assert html =~ "Pending approval"
     end
+
+    test "plot-path-to-beacon link pre-populates tx_power + gain to match beacon EIRP",
+         %{conn: conn} do
+      # power_mw: 10_000 → EIRP 40 dBm → default-gain 30 dBi → tx 10 dBm
+      beacon = approved_beacon_fixture(user_fixture(), power_mw: 10_000.0)
+      {:ok, _view, html} = live(conn, ~p"/beacons/#{beacon}")
+
+      assert html =~ "tx_power_dbm=10.0"
+      assert html =~ "src_gain_dbi=30.0"
+    end
+
+    test "low-EIRP beacon clamps tx_power to 0 dBm floor and lowers gain", %{conn: conn} do
+      # power_mw: 100 → EIRP 20 dBm → tx 0 dBm (clamped), gain 20 dBi
+      beacon = approved_beacon_fixture(user_fixture(), power_mw: 100.0, callsign: "W5LOW")
+      {:ok, _view, html} = live(conn, ~p"/beacons/#{beacon}")
+
+      assert html =~ "tx_power_dbm=0.0"
+      assert html =~ "src_gain_dbi=20.0"
+    end
   end
 
   describe "Form" do