The Cayley–Dickson Ladder 3: Termination of Loss and Proliferation of Degeneration—The Machine-Checked Structure of the Cayley–Dickson Ladder

The termination problem for the Cayley–Dickson ladder has long remained atthe narrative level of “one property is lost at each step”; its difficulty lies in thefact that the word “loss” lacks a testable formal object. This paper strictly relativizes the termination statement to an explicitly given property domain P: Pconsists of seven equational identities, namely self-conjugacy, commutativity, associativity, alternativity, flexibility, power-associativity, and multiplicativity of thenorm. Under this convention, the set of death levels is precisely determined tobe {1,2,3,4}: self-conjugacy dies at level 1, commutativity at level 2, associativity at level 3, and alternativity and multiplicativity of the norm both die at level4; flexibility and power-associativity extend indefinitely along the whole ladder bySchafer’s theorem. The fourth level simultaneously accommodates the death of twoproperties, so the reading “exactly one property per level” is inconsistent with thecounting of the set of death levels; termination must be stated in terms of the levelset rather than the number of properties. On the degeneration side, the existenceof zero divisors is registered separately as the quasi-identity Q, with d(Q) = 4, anddoes not enter the death-level count of P; the embedding ι(x) = (x,0) and thenew-half lift ρ(u) = (0,u) together give a proper cross-generational nesting, underinclusion, of the set of nonzero zero divisors; same-half and cross-half combinationsof the two give four embedding lifts, so that the addressed count Z(n) of mutually annihilating pairs of first-two sparse imaginary units is strictly increasing andunbounded, explicitly Z(n) ≥ 4n−4Z(4) for n ≥ 4. The addressed count satisfiesZ(n +1) = 4Z(n)+V(n), where V(n) is the count of nonzero associator triples;the newly added part of degenerate proliferation is precisely driven by the residualassociator defect on the equational side, and the existence and disjointness of the4Z(n) lift term are verified independently in this paper. All proliferation statementsare restricted to the addressed-count object and do not assert anything about thecardinality of the full set of zero divisors; whether there are new death levels outsideP is registered as an open problem and is not mixed into proved conclusions.