Arc 4: The Primordial A Three-Part Descent Through Substrate, Orchard, and Identification

Developmental Geometry Arc 4 — Substrate Identification This volume identifies the structural substrate of the Developmental Geometry (DG) program. Building on the generative framework developed in Arc 3 (imbalance as primitive, the generative triple GG G, and the Universality Theorem), Arc 4 asks what specific algebraic and geometric structure carries the framework's content. The main result identifies the substrate as the symmetric space SO(4,2)/SO(4)×SO(2)SO(4,2) / SO(4) \times SO(2) SO(4,2)/SO(4)×SO(2) in non-compact form, with su(2)L⊕su(2)R\mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R su(2)L⊕su(2)R isotropy decomposition and B2B_2 B2 long-root structure. The identification proceeds from five axioms codifying: a warped-product developmental metric, cone geometry with structurally unreachable boundary, two-state perpendicular structure, canonical arithmetic alignment, and closure of admissibility under the generative cascade. Principal results. Substrate Identification Theorem: under the five axioms, the substrate is uniquely identified up to the standard equivalences. The isotropy splitting into two commuting su(2)\mathfrak{su}(2) su(2) factors as a structural consequence of the two-state perpendicular condition. The arithmetic alignment as the null-line family Farith=P1\mathcal{F}_{\text{arith}} = P_1 Farith=P1, singled out among the four admissible alignment families. The classification of alignment families, parabolic conjugacy classes, isotropy orbits, and dominant eigenvalue structure, all under the axiomatic conditions. Placement in the program. Arc 4 is the substrate-identification monograph. Arc 5 (forthcoming) identifies the pre-substrate — the structure that forces the substrate — and adds the recursion condition (Axiom 6). Arc 3 provides the generative framework (GG G and UU U) that Arc 4's substrate realizes. Arc 2 (three volumes) provides the analytic framework. Arc 1 develops the underlying metric and analytic foundations. Arc 0 establishes the primitive bifurcation and perpendicularity necessity. Related companion works published separately include derivations of specific substrate consequences: PaperE (spectral existence on the developmental warped product), PaperF (the Lorentz-form time invariant and structural censorship), PaperG (turbulence correspondence), Intergenesis (axiomatic derivation of the timing functional), and others on Zenodo. Intended audience. Mathematicians and mathematical physicists interested in symmetric-space classifications, structural derivations of foundational physical frameworks from non-physical axioms, and the geometric structure of imbalance-driven cascades. Readers may enter the program at Arc 4 if familiar with the SO(4,2)SO(4,2) SO(4,2) conformal structure and symmetric space theory; readers preferring a ground-up exposition should begin with Arc 0 and work forward.