Arc 6: The Wild Frontier

Arc 6: The Wild Frontier The first book of the Developmental Geometry program's outward motion. Arc 6 articulates the wild regime — the developmental mode in which prior development leaves residue that constrains subsequent development — as the structural complement to the tame regime developed across Arcs 1–5. Together with Arc 7, this volume opens the program's second arc-pair, treating the tame–wild duality on the substrate as the simplest and most fundamental developmental classification. The volume is organized in three Parts. Part I introduces the wild regime's primitive objects: spirals (curves with self-repeating curvature signature under recursive scaling), lineages, families, and wild corridors as foliated regions of admissible development. Part II articulates the two principles governing wild-regime evolution — Developmental Memory (DM) and Contact-Induced Memory (CM) — and proves their consistency. Part III develops the structural consequences: shadow accumulation as a scalar field on the developmental manifold, the collapse boundary as the locus of saturation, the inverse cone dual to Arc 6's light cone, the wild straight line, and the Wild Relaxation Law governing the three-phase cycle of accumulation, saturation, and release. Central technical content includes: A tubular construction for the curvature residue, supplying a Lipschitz-continuous shadow field on the developmental manifold whose level sets foliate the finite-shadow region. The Monotonicity Theorem (Chapter 4) and Wild Relaxation Theorem (Chapter 11) as structural consequences of the prior commitments. The Consistency Theorem (Chapter 6) establishing that DM and CM together preserve corridor structure across wild-regime evolution. A formalization of the inverse cone via a non-trivial trapping condition, dual to the light cone of free propagation. The volume is the first physics-relevant articulation of the wild regime as a formal mathematical object. The substrate identification follows Arc 4's compact rank-2 Riemannian symmetric space SO(6)/(SO(4)×SO(2)); the pre-substrate context is Arc 5. Heat is introduced as a named wild-regime invariant with its own propagation law (the Heat-Alignment Law), with its promotion to triadic primitive structure deferred to Arc 7. Six open mechanisms are registered honestly at minor severity, covering quantitative refinements (spiral parameter bounds, ε-dependence of the curvature residue, corridor coherence threshold) and structural identifications (DM/CM constructive non-emptiness, attractor signature formal identification, heat tubular neighborhood width). These are deferred to a future analytic treatment without affecting the volume's structural arc.