Arc 8 — The Expanse

Arc 8 of the Developmental Geometry program articulates the wild-regime expanse: the static-geometry structure of the admissible region within the depth hierarchy, with empirical realizations spanning atomic, hadronic, and orbital alignments. The arc establishes the wild-regime operator on a Hilbert space of curvature-content sequences, derives the period-3 envelope as a structural skeleton, and characterizes the admissible range [7,13][7, 13] [7,13] via the dual-condition characterization of the prime-family hierarchy. Cone-geometry corrections produce the band-width layer; the prime-family depth hierarchy produces the boundary layer; together these structures constitute the admissibility geometry of the wild regime. Cross-arc corroboration registers the equation-form alignment between Arc 3's shell-capacity arithmetic (Cn=2n2C_n = 2n^2 Cn=2n2) and Arc 8's admissibility geometry. The reconstructed radius rule rn=(Cn/2)ur_n = (C_n / 2) u rn=(Cn/2)u recovers hydrogen orbital radii at atomic register under u=a0u = a_0 u=a0 and identifies the same structural arithmetic at orbital register. Cross-system orbital fits across six planetary systems support the framework at sub-percent precision in formation-preserved bodies. Empirical content includes: Hadron mass cluster correspondence with admissibility boundaries and interior anchors Atomic-register exact match under shell-capacity arithmetic Cross-system cold-giant shell match at sub-0.1% residual across three independent planetary systems (Solar Jupiter, 55 Cnc d, Kepler-90 g/h) Joint Mercury+Jupiter anchor at 0.05% Solar internal precision Six-system survey identifying observed organizational regimes Within-Solar-system angular momentum coupling pattern: Ltotal⋅∣cos⁡(θ)∣L_{\text{total}} \cdot |\cos(\theta)| Ltotal⋅∣cos(θ)∣ produces ~1184× cluster separation between bodies with negative and positive shell-residual signs The arc closes with explicit registration of open structural questions: the Path B correspondence, structural derivation of the mass-timing relation, the shell-capacity radius rule, two-body curvature reconciliation, body-flow angular momentum coupling, and structural unification of the lepton and hadron readings under Axiom 6. Arc 8 takes the wild-regime structure inherited from Arcs 6-7 (depth filtration, period-3 envelope, prime-family hierarchy) and identifies its admissible region with structural and empirical content. It is the static-geometry arc; dynamical content (contact regime, body-flow coupling derivation) is reserved for Arc 9.