Extension of Axiom A1 to an Asymptotically de Sitter Bulk: A Conditional Analysis

Axiom A1 of the COCM formal foundations postulates a hierarchical sequence of concentric ring branes embedded in a five-dimensional AdS bulk with a standard decreasing warp factor. The observable universe has a positive cosmological constant, suggesting the physical bulk may be asymptotically de Sitter rather than AdS. This note examines whether A1 can be extended to this case, and shows that A1 can extend only locally and conditionally in an asymptotically de Sitter bulk: the de Sitter warp factor reaches a cosmological horizon at a finite proper distance y_H, which imposes an upper bound on the inter-brane separations that A1 assumes can be freely chosen. If Λ₅ is sufficiently small relative to the brane tensions, the geometry near C₀ is quasi-AdS and A1 extends locally within a domain bounded by y_H. The genuinely open problem is whether this local extension is sufficient to preserve the full theorem chain that A1 supports in the global AdS case.