Support-Class Necessity I: Convexity, Endpoint Antipodes, and Balanced Segment Allocation

Round 03 compressed the problem to 'does the optimum lie in the support-tangent class?'; this paper attacks that necessity directly. Finite-M convexity and the endpoint-strip reduction need no new proof and are imported from Adhikari–Pitman 1989 (two external gates); inside the strip the endpoint-antipode identity is automatic for a convex upper chain; the interior support-slack lemma shows interior side edges must be tangent to the unit circle about the opposite endpoint; and in the reduced support-tangent class the one-side reduced length is strictly jointly convex in (q, a), so the left and right segment budgets must be as balanced as possible (symmetric for even n, an asymmetric mirror pair for odd n). The remaining main gates are boundary saturation and cell-boundary exclusion.