Question: Which monarch of the Mexican Empire was the 19th-century castle on the Gulf of Trieste Miramare Castle built for?

full_context: Document 1: Archduke Franz Ferdinand of Austria: Franz Ferdinand Carl Ludwig Joseph Maria (18 December 1863 – 28 June 1914) was an Archduke of Austria-Este, Austro-Hungarian and Royal Prince of Hungary and of Bohemia and, from 1896 until his death, heir presumptive to the Austro-Hungarian throne.
Document 2: Porfirio Díaz: José de la Cruz Porfirio Díaz Mori (] ; 15 September 1830 – 2 July 1915) was a Mexican general and politician who served seven terms as President of Mexico, a total of three and a half decades from 1876 to 1880 and from 1884 to 1911. A veteran of the War of the Reform (1858–60) and the French intervention in Mexico (1862–67), Díaz rose to the rank of General, leading republican troops against the French-imposed rule of Emperor Maximilian. Seizing power in a coup in 1876, Díaz and his allies, a group of technocrats known as "Científicos", ruled Mexico for the next thirty-five years, a period known as the "Porfiriato".
Document 3: Maximilian I of Mexico: Maximilian (Spanish: "Maximiliano"; born "Ferdinand Maximilian Joseph"; 6 July 1832 – 19 June 1867) was the only monarch of the Second Mexican Empire. He was a younger brother of the Austrian emperor Francis Joseph I. After a distinguished career in the Austrian Navy, he accepted an offer by Napoleon III of France to rule Mexico. France (along with the United Kingdom and Spain, who both withdrew the following year after negotiating agreements with Mexico's democratic government) had invaded Mexico in the winter of 1861, as part of the War of the French Intervention. Seeking to legitimize French rule in the Americas, Napoleon III invited Maximilian to establish a new Mexican monarchy for him. With the support of the French army, and a group of conservative Mexican monarchists hostile to the liberal administration of new Mexican President Benito Juárez, Maximilian traveled to Mexico. Once there, he declared himself Emperor of Mexico on 10 April 1864.
Document 4: Miramare Castle: Miramare Castle (Italian: "Castello di Miramare" ; German: "Schloss Miramar" ; Slovene: "Grad Miramar" ) is a 19th-century castle on the Gulf of Trieste near Trieste, northeastern Italy. It was built from 1856 to 1860 for Austrian Archduke Ferdinand Maximilian and his wife, Charlotte of Belgium, later Emperor Maximilian I and Empress Carlota of Mexico, based on a design by Carl Junker.
Document 5: Ferdinand, Duke of Breisgau: Archduke Ferdinand Karl of Austria-Este (1 June 1754 – 24 December 1806) was a son of Holy Roman Emperor Franz I and Maria Theresa of Austria. He was the founder of the House of Austria-Este and Governor of the Duchy of Milan between 1765 and 1796. He was also designated as the heir to the Duchy of Modena and Reggio, but he never reigned, owing to the Napoleonic Wars.
Document 6: Maximilian I, Holy Roman Emperor: Maximilian I (22 March 1459 – 12 January 1519) was King of the Romans (also known as King of the Germans) from 1486 and Holy Roman Emperor from 1493 until his death, though he was never crowned by the Pope, as the journey to Rome was always too risky. He was the son of Frederick III, Holy Roman Emperor, and Eleanor of Portugal. He ruled jointly with his father for the last ten years of his father's reign, from c. 1483 to 1493.
Document 7: Carl Junker: Carl Junker (18 June 1827 – 17 May 1882) was an Austrian engineer and architect. His construction projects include Miramare Castle in Trieste and the First Vienna Mountain Spring Pipeline.
Document 8: Miraz's Castle: The Castle of Caspian, also known as Miraz's Castle, was home to the early Telmarine Dynasty (the "Caspians") of the country of Narnia. The castle was built by Caspian VI and was home to him and his successors, including Caspian VII and Caspian VIII. The last permanent ruler of the palace by the name of "Caspian" was Caspian IX, who was murdered and succeeded by his brother Miraz. After Miraz was murdered at the Second Battle of Beruna, Caspian X may have lived in it briefly as king before rebuilding Cair Paravel and making it his new palace.
Document 9: Barcola: Barcola is a maritime neighbourhood of Trieste, Italy. It is a popular tourist place with beaches and long promenade walkways, near to the Habsburg-established Miramare Castle.
Document 10: Military Order of Max Joseph: The Military Order of Max Joseph (German: "Militär-Max-Joseph-Orden") was the highest military order of the Kingdom of Bavaria. It was founded on 1 January 1806 by Maximilian I Joseph of Bavaria, the first king of Bavaria. The order came in three classes:
Document 11: Juarez and Maximillian: Juarez and Maximillian (Spanish:Juárez y Maximiliano) is a 1934 Mexican historical drama film directed by Miguel Contreras Torres and Raphael J. Sevilla. The film is set during the French intervention in Mexico during the 1860s, and features the battle between Maximilian I of Mexico and Benito Juárez, a theme used in the later 1939 American film "Juarez". It was one of the few major commercial successes for the Mexican film industry in the early sound era, before the beginning of the Golden Age of Mexican cinema.
Document 12: Archduke Maximilian of Austria-Este: Archduke Maximilian Joseph of Austria-Este (July 14, 1782 – June 1, 1863), the fourth son of Archduke Ferdinand of Austria-Este and younger brother of Francis IV, Duke of Modena. He was grand master of the Teutonic Knights from 1835 to 1863.
Document 13: Archduke Franz Karl of Austria: Archduke Franz Karl Joseph of Austria (17 December 1802 – 8 March 1878) from the House of Habsburg was father of two emperors (Franz Joseph I of Austria and Maximilian I of Mexico) as well as the grandfather of Archduke Franz Ferdinand of Austria, whose assassination sparked the hostilities that led to the outbreak of World War I, and the great-grandfather of the last Habsburg emperor Karl I.
Document 14: Sforza Castle: Sforza Castle (Italian: "Castello Sforzesco" ) is in Milan, northern Italy. It was built in the 15th century by Francesco Sforza, Duke of Milan, on the remains of a 14th-century fortification. Later renovated and enlarged, in the 16th and 17th centuries it was one of the largest citadels in Europe. Extensively rebuilt by Luca Beltrami in 1891–1905, it now houses several of the city's museums and art collections.
Document 15: Ferdinand I, Holy Roman Emperor: Ferdinand I (Spanish: "Fernando I" ) (10 March 1503 – 25 July 1564) was Holy Roman Emperor from 1558, king of Bohemia and Hungary from 1526, and king of Croatia from 1527 until his death. Before his accession, he ruled the Austrian hereditary lands of the Habsburgs in the name of his elder brother, Charles V, Holy Roman Emperor. Also, he often served as Charles' representative in Germany and developed useful relationships with German princes.
ground_truth: Emperor Maximilian I