Aigner's Three Monotonicities from Fibonacci Matrices and Six Snake Pieces

Starting from the established fact that Markov numbers count perfect matchings of their Markov snake graphs, this six-page paper gives a self-contained proof of Aigner's fixed-numerator, fixed-sum and fixed-denominator orderings. The author's Fibonacci matrices establish normalization and fixed-numerator monotonicity; his six-piece decomposition establishes the remaining fixed-sum comparison and fixed-denominator monotonicity. The normalization begins with (AC)_11 - (B^2)_11 = 2 for consecutive odd-indexed Fibonacci blocks. The fixed-sum example is illustrated in Figure 2, and the three ordering proofs have parallel subsection headings. The matching correspondence is credited to Musiker's 27 June 2002 memorandum and the Itsara–Musiker–Propp–Viana manuscript of 10 January 2003; earlier proofs of Aigner's orderings are cited separately. Included are the six-page PDF and editable LaTeX source, figures, and finite-check scripts and reports. Funding: No funding. Generative-AI assistance in proof development, exact checking, reference retrieval and typesetting is disclosed in the manuscript; the author directed the research and is responsible for the work.