The Cayley–Dickson Ladder 1: Four-Step Loss and Termination

The Cayley–Dickson construction starts from the real numbers and doubles successively through the complex numbers, quaternions, and octonions to the sedenionsand higher-dimensional algebras. The changes in algebraic properties at each stephave long appeared in the literature as scattered case studies, and a ladder-stylestatement based on a unified generation rule, locatable addresses, and decidableranges is lacking. This paper takes three inputs—the multiplication rule, the conjugation rule, and the norm rule—as a single recursive source, and generates from thereal base all multiplication tables, conjugation tables, norm tables, and zero-divisorsets up to the 64-dimensional algebra; dimension-condition branching is forbiddeninside the generator, and seven property verifiers are written independently anddo not share decision subroutines. On this basis, we define the property domainP, whose members are seven equational identities: self-conjugacy, commutativity,associativity, alternativity, flexibility, power-associativity, and multiplicativity ofthe norm. The death hierarchy characterizes the smallest algebra level at whicheach item in P first fails, yielding finite death levels exactly 1,2,3,4, where thefourth level simultaneously contains alternativity and multiplicativity of the norm;flexibility and power-associativity are borne by Schafer’s theorem as universally surviving and do not enter the finite death hierarchy. The existence of zero divisors istreated as a structural degeneracy marker independent of P: its nesting is carriedby the proved lemma that embeddings preserve operations; finite-level canonicalcounting belongs to the data layer; and proliferation without end is registered asa framework-internal proposition, not elevated to a universal construction. Identity scanning outside P with degree at most 4 and level at most 6 is given as abounded computational proposition, and its universal version is listed as an openproblem. This paper also systematizes the registration status of two evidence tracks:machine-checked assertions are registered in four types of registration tables withthree states [pass/fail/withdraw], and the verification of applicability conditions onthe classical track is included in the same registration system; during the debt period, the corresponding statements remain in conditional form. The terminationstatement is therefore limited to termination relative to P and infinite extension atthe Schafer-theorem layer, rather than a completeness assertion about all equationalproperties or all degeneracy processes.