φ as a Cross-Branch Survival Filter: A Necessity Recursion Approach to Observer Selection in Branching Universes

We propose an exploratory framework extending Fibonacci Causal Loop Theory (FCLT) to branching universe selection. Under necessity recursion S(n) = S(n-1) + S(n-2), φ ≈ 1.618 emerges as the only stable attractor across all sufficiently deep necessity systems. We introduce the Branch Survival Function: P(B) = e^(-k₀·D(B)·δ(B)) where δ(B) measures the φ-deviation of branch B, k₀ is the φ-Persistence Coefficient, and D(B) is necessity depth. We propose that branches converging toward φ-structured organization persist and complexify, while high-deviation branches collapse toward lower necessity depth. This provides a mathematically grounded selection mechanism independent of anthropic circular reasoning. Connections to the Born Rule derivation (P15), Davis Attractor stability (P07), and decision-branch framework (P47) are developed and unified into a four-level FCLT hierarchy. The Clark Hamiltonian Filter (Davis & Clark, 2026; Clark, 2026a-d) is identified as the complementary first stage of a complete two-stage selection framework. This paper is designated exploratory pending derivation of falsifiable predictions. Four testable predictions are formally pre-registered.