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

The Cayley–Dickson construction starts from the real numbers and doubles suc
cessively through the complex numbers, quaternions, and octonions to the sedenions
and higher-dimensional algebras. The changes in algebraic properties at each step
have long appeared in the literature as scattered case studies, and a ladder-style
statement based on a unified generation rule, locatable addresses, and decidable
ranges is lacking. This paper takes three inputs—the multiplication rule, the conju
gation rule, and the norm rule—as a single recursive source, and generates from the
real base all multiplication tables, conjugation tables, norm tables, and zero-divisor
sets up to the 64-dimensional algebra; dimension-condition branching is forbidden
inside the generator, and seven property verifiers are written independently and
do not share decision subroutines. On this basis, we define the property domain
P, whose members are seven equational identities: self-conjugacy, commutativity,
associativity, alternativity, flexibility, power-associativity, and multiplicativity of
the norm. The death hierarchy characterizes the smallest algebra level at which
each item in P first fails, yielding finite death levels exactly 1,2,3,4, where the
fourth level simultaneously contains alternativity and multiplicativity of the norm;
flexibility and power-associativity are borne by Schafer’s theorem as universally sur
viving and do not enter the finite death hierarchy. The existence of zero divisors is
treated as a structural degeneracy marker independent of P: its nesting is carried
by the proved lemma that embeddings preserve operations; finite-level canonical
counting belongs to the data layer; and proliferation without end is registered as
a framework-internal proposition, not elevated to a universal construction. Iden
tity scanning outside P with degree at most 4 and level at most 6 is given as a
bounded computational proposition, and its universal version is listed as an open
problem. This paper also systematizes the registration status of two evidence tracks:
machine-checked assertions are registered in four types of registration tables with
three states [pass/fail/withdraw], and the verification of applicability conditions on
the classical track is included in the same registration system; during the debt pe
riod, the corresponding statements remain in conditional form. The termination
statement is therefore limited to termination relative to P and infinite extension at
the Schafer-theorem layer, rather than a completeness assertion about all equational
properties or all degeneracy processes.