Re-derivation of r* and Verification of A5 Locality under the A1-dS Extension: Conditional Closure of Problem PH

This note completes the compatibility assessment of the P4 theorem chain under A1-dS by computing r* explicitly and verifying the locality of A5. We show that r* is finite, positive, and reduces to the AdS value as Lambda_5 -> 0. We also confirm that Axiom A5 depends only on brane-position data. As a result, T4 is confirmed under the domain restriction y_RN < y_H (proof completed), and T7 is confirmed DOMAIN. All nine theorems of P4 survive the A1-dS extension: five are ROBUST and four are DOMAIN. Problem PH is conditionally closed under NP1-NP3.