[potiuper]

Iota appears to be the same as the "X" combinator as found by Jeroen Fokker: The Systematic Construction of a One-Combinator Basis (1989) that amounts to concatenating/applying (material) implication's reduction & distributivity axioms ("S" & "K") in addition to first binding another variable "x", which is "f" in the Iota definition. Shorter than "x" locally complete & consistent (indecisive) single implication axioms exist as one found by Meredith and listed by the now deceased Dolph Ulrich: https://web.ics.purdue.edu/~dulrich/C-pure-intuitionism-page... See also: https://math.stackexchange.com/questions/153163/what-is-the-... The shortest single implication axiom appears to be an open question (though translating to implication the McCune/Fitelson/Wos/Wolfram shortest 6 NAND axiom work https://en.wikipedia.org/wiki/Minimal_axioms_for_Boolean_alg... seems to be a proof for 3 variables) along with if distributivity is the shortest local completeness (weakening) axiom (see Robbins axiom & associativity & commutativity 3-basis) in addition to having to analyze Meredith's & others on Ulrich's list for self-interpreter lengths. But, a self-interpreter seems unlikely to be shorter compared to lambda calculus given the pattern of decreasing self-interpreter length with increasing complexity of the underlying axiom system.