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Theoretical analysis demonstrates arithmetic thermodynamics in Hurwitz quaternion monoids, indicating a partition function with critical exponent two.
The Hurwitz quaternion order provides a genuinely non-commutative right Ore, right LCM monoid whose Nica-Toeplitz system has arithmetic thermodynamics, with partition function zeta(beta) zeta(beta-1) and critical exponent 2. Conditions (C1)-(C4) for a non-injective arithmetic Bost-Connes factor are formulated, the unit-group exclusion (Remark 1.5: an infinite unit group makes the partition function diverge identically, and a finitely generated monoid has constant obstruction group) narrows the search, and (C4), non-amenability of the measured groupoid, is posed; it is settled negatively in Paper XXVIII of the series.
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Ruqing Chen (2026) studied this question.
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