Wash your hands, then clean and sanitize the counter, cutting board, knife and other utensils. Wash the 1-quart heat-safe jar and its acid-resistant lid with hot soapy water, then rinse thoroughly.

Place the washed jar on a rack in a deep pot and cover it completely with water. Gently simmer for 10 minutes, then lift it out carefully and keep it warm. Prepare the lid according to its manufacturer's directions.

Rinse the produce under clean running water before peeling and slicing. Slice the onion, ginger, horseradish, garlic and jalapeño very thinly, measure the listed quantities, and place them in the warm jar. The combined produce should occupy 1 cup; do not pack in additional solids.

Heat the vinegar in the saucepan to 190–195°F (about 88–91°C). Pour it over the sliced produce. Use the probe to verify that all the liquid and the center of the added solids stay above 165°F (74°C) for at least 1 minute 12 seconds. If the temperature is too low, place the jar on a rack in a gently simmering water bath with the water below the jar rim and heat until that measured holding target is met. Use jar-lifting tongs and take care with the hot glass; do not skip the temperature check.

Allow the jar to cool enough to handle safely, close the lid, and refrigerate within 2 hours. Keep the infusion at 40°F (4°C) or colder for 10–30 days. Swirl the closed jar occasionally; the solids should stay covered by vinegar.

After 10 days, check the infusion for spoilage before sampling a few drops with food. When the flavor suits you, strain through a coffee filter into a freshly cleaned and sterilized glass bottle. Discard all the solids; do not add fresh garnish to the stored vinegar.

Close the bottle, date it from the original preparation day and keep refrigerated. Use within 3 months of that date. Discard without tasting if mold, unexpected cloudiness, bubbling or sliminess appears. This recipe is not intended for home canning or room-temperature storage.