f(h) ≈ f(h*) + J(h−h*) — the lens is the tangent plane; curvature is what it throws away
tier: lit · math witnessed in node (J residual 5.2e-10, quad coeff 15.089 vs 15.117) · 2026-07-09 · bridge-burners
2D — TANGENT VS TRUTH · ERROR SCALING · κ DIAL
true readout p·f(h*+tu)
jacobian tangent p·(f₀+tJu)
curvature gap
direction u sweeps the (u₁,u₂) plane automatically
3D — MANIFOLD SLICE VS TANGENT PLANE (SOFT-GL)
true surface z = p·f(h*+a·u₁+b·u₂)
tangent plane from J
anchor h*
the daylight between them is the curvature