The Ladder of Time E2: Causal Fragments and Failed Universes — The Uninhabitability Theorem of the Sedenion Level

Starting from the algebraic rigidity of the Hurwitz classification theorem for
normed division algebras, this paper systematically demonstrates the physical con
sequences of crossing the octonion threshold into the sedenion level. By construct
ing the Hurwitz-time functor, the causal structure of event space is rigorously
mapped onto the axiom retention pattern of algebraic structures, thereby establish
ing a complete chain of deduction from algebraic pathology to physical causal frag
mentation. At the algebraic level, the sign flip introduced in the fourth step of the
Cayley-Dickson construction inevitably leads to the high-dimensional emergence of
a zero-divisor space, the discretization transition of the automorphism group from
a connected exceptional Lie group to a product structure, and the irreconcilable
contradiction between locally associative subalgebras and global divisibility. At the
causal level, the coexistence of zero divisors and non-associativity forces the event
space to split into multiple mutually incompatible local coarse-graining protocols,
and cross-protocol information annihilation becomes an operationally unavoidable
mechanism of causal rupture. At the operational level, the self-referential repair
operator admits no self-consistent solution on such fragmented states, and the re
sulting upgrade deadlock renders the sedenion level neither internally repairable nor
externally upgradable. The paper further argues for the inevitability of the observ
able universe stably residing at the octonion level: the unstable traversal scenario
is ruled out by the upgrade deadlock theorem, and the necessary-residence scenario
remains the only self-consistent cosmological explanation. The ultimate boundary
of temporal level upgrades thus exhibits a dual rigidity—thermodynamic capping
is locked at the energy level by Landauer’s principle, while algebraic pathology is
locked at the structural level by the Hurwitz theorem.