This paper presents the G∞-RCD(QAP) framework for deriving the cosmological constant from first principles. The framework development proceeded through iterative refinement, with the final formulation replacing two initial working assumptions: (1) the QAP background geometry changes from the round 4-sphere S4 to a branching tree with depth parameter n and natural branching number b = e; (2) the cosmological constant is shown to be measured not by the variance σ²vac of the WKB action distribution but by the total information depth Dtotal = S(Ξ* ∥ Ξvac), the relative entropy (KL divergence) between the G∞ fixed-point measure Ξ* and the vacuum measure Ξvac. Crucially, the Peters formula ρΛ = ρQFT · exp(−σ²vac/2) is derived from first principles (Section 5A) via topological non-ergodicity of the dendritic QAP graph, eliminating the need to import it from ergodicity economics. The main theorem (Theorem X — the Information-Depth Peters Formula) proves: ρΛ = ρQFT · exp(−Dtotal) with Dtotal = dim(Sym²(ℝ³)) × |ζ′SM(0; leaf S4)| × n²max / 2 = 6 × (11/180) × (39.34)²/2 = 283.75 nats, giving ρΛ/ρQFT ≈ 10−123. A complementary variance-based formulation provides an equivalent perspective: σ²vac = 2Dtotal = 567.5 with ρΛ = ρQFT · exp(−σ²vac/2). The factor of 1/2 between Dtotal and σ²vac is derived (not assumed) from the Gaussian path-integral structure of the branching QAP. GAP FP-2 (the log²-running theorem) is fully resolved: the log²(MPl/TEW) factor arises from the linear growth In = (11/180)·n of the per-level information density with depth n, giving ∑n=1nmax In = (11/180)·n²max/2. The branching QAP with b = e is identified with Euclidean AdS5 bulk, and the per-level information density 11/180 is identified as the Weyl anomaly coefficient of the boundary CFT4 on S4. Natural branching b = e is shown to be the unique value consistent with the observed cosmological constant.