Hyperoperations of real rank: a canonical continuum from algebraic rigidity
Frédéric Morneau-Guérin, Tony Tsz Hong Ip · R-libre (Université Téluq) · 2026
The hyperoperations -- addition, multiplication, exponentiation, tetration, and so on -- form a hierarchy in which each member iterates the previous one. We construct a family (H_r) indexed by a real rank r, passing through every member of the hierarchy and jointly continuous in (r,m,n) on a natural rank-dependent domain. The continuity theorem is sharp: the only point of failure on the domain is (3,0,0), an obstruction already present in the rank-3 operation m^n. The construction is not an arbitrary interpolation. Between addition and multiplication, the operations are characterised axiomatically by continuity, associativity, strict monotonicity, the existence of a neutral element, homogeneity of the associated mean, and the calibration H_r(2,2)=4. Between multiplication and exponentiation they are forced by the exponent law, and a previously proposed construction by successive halvings is shown to converge to the same family. Above rank 3, a rigidity theorem singles out the exponential seed among Hooshmand-type extensions satisfying a rank-independent seed law. We establish sharp monotonicity regimes in the rank: the family is monotone on (-3,3] for all m,n greater or eqal to 2, and for every real height x greater or equal to 1 it is monotone on [2,infinity) precisely when the base is at least 4log(2). We also continue the hierarchy below rank 1, where the calibration produces an asymptote at rank -3 and the operations degenerate to max(m,n,4). The resulting family is canonical in the rank rather than in the height. Above rank 3 it is of Hooshmand type and generally has corners in the height, whereas analytic Abel-theoretic extensions provide a different notion of canonicity. We discuss the relation between these two approaches.