The Ladder of Stability I: The Double Necessity of Failure for Sedenion Algebras

Within a conditional methodological framework and using the Cayley–Dickson
algebraic hierarchy as a thread, this paper demonstrates the double necessity for
sedenion algebras to lose normed divisibility. First, starting from the minimal op
erational requirements of a self-consistent event space, we prove that informational
distinguishability, causal transitivity, and elementary decomposability jointly force
the event space to constitute a finite-dimensional normed division algebra; Hur
witz’s theorem then strictly locks the maximum viable dimension to the octonions.
Second, by introducing a dynamical analysis of the co-constitution of emergence
and stability, we define two independently measurable sequences — emergent ef
ficacy and stability cost — and prove that along the directed evolution of the
Cayley–Dickson tower, emergent efficacy peaks at the octonions and vanishes at
the sedenions, while stability cost undergoes an irreversible jump to infinity at the
same level, thereby breaking the co-constitution relation. The two pathways con
verge precisely at the octonion level, revealing that the metastably compressed state
of sedenions is both a necessary consequence of algebraic rigidity and a necessary
endpoint of hierarchical evolutionary dynamics. Under the given meta-constraints,
the paper excludes competing algebraic paths and establishes counterfactual ro
bustness of the conclusion based on the path-independence of Hurwitz’s theorem.
Appendices provide complete formal proofs, explicit constructions of zero divisors,
and information-thermodynamic anchoring, making the chain of reasoning from
algebraic boundaries to physical limits logically self-consistent and closed.