Characteristic Cone Geometry and Calm-Regime Conditions for Poisson Gravity in a Constrained S3 Field

We derive emergent spacetime geometry from a constrained field Φ: ℝ³⁺¹ → S³. The characteristic metric g_char (from principal symbol analysis) governs wave propagation but has short-range tails (r⁻⁶), ruling out Newtonian gravity. A separate gravitational metric g_grav[Φ] is formulated in the calm regime with stretch potential τ. Using Φ-only knot energy density ρ_knot from Skyrme invariants, we obtain ∇²τ = 4πG₀/c⁴ ρ_knot, yielding 1/r Newtonian tails. This distinguishes optical geometry (local waves) from gravitational geometry (global matter), providing falsifiable tests for emergence within Postulate 1 (Φ-only ontology).