The Curvature–Prime Depth Hierarchy

We construct and analyze a depth hierarchy on the primes, generated by a recursive packet-sum rule with a finite-tilt condition. Beginning from a single primitive P₁ = {3}, each successive depth class P_{n+1} consists of primes expressible as sums of exactly three packets from P_n, subject to a condition excluding the pure-container primitive. We verify the hierarchy explicitly through P₈ (primes up to 15,000) and prove two structural results. First, the parity theorem: for any depth class consisting of odd primes, the minimum admissible generator count is k = 3, forced by parity alone. This implies that the ratio min P_{n+1} / min P_n converges to 3 as n → ∞. The convergence factor is determined by parity, independent of the choice of tilt primitive. Second, the DG′ collapse theorem: swapping the tilt primitive from 3 to 5 causes the hierarchy to collapse immediately at P₂, establishing that the assignment of 3 as tilt is not symmetric under role permutation. The finite-tilt condition itself blocks every bootstrap path under role-swap; only tilt = 3 yields a viable hierarchy. We establish a structural characterization of two distinguished primes within P₂: a dual condition (substrate-only decomposition together with appearance as a doubled component in deeper P₂ elements) picks out exactly {7, 13} from P₂. This characterization underwrites the boundary structure of the admissible region for hadronic curvature in Arc 8 (10.5281/zenodo.20030252), where it identifies the boundary primes of the wild-regime admissible range. The hierarchy exhibits structural parallels to the developmental cascade in Developmental Geometry (DG), including non-termination, unbounded depth increase, and arithmetic-driven class sparsity. We state the Path B correspondence as a precise open conjecture: that a DG-internal mapping exists identifying the arithmetic and geometric structures. No such identification is claimed here; the paper presents the arithmetic object independently. The arithmetic structure stands self-contained whether or not the geometric correspondence is eventually established.