Solve minimum-cost shipping plan and prove optimality

Solve a self-contained operations-research problem and deliver a decision report, not an application. Three warehouses A,B,C have supply 7,9,8 whole crates; shops X,Y,Z demand 8,6,10. Per-crate transport costs in rows A,B,C and columns X,Y,Z are [[4,6,8],[5,3,7],[6,5,2]]. Route A->Z is unavailable. Route B->Y can carry at most 4 crates. All other routes have no capacity beyond supply/demand. Find an integer minimum-cost shipment matrix, prove optimality (a rigorous lower-bound certificate or complete bounded enumeration), and determine the new optimum when B->Y capacity increases from 4 to 5. State whether each optimum is unique. Deliver the numerical answer and explanation directly in the final response, plus report.md and solution.json containing both scenarios and a small reproducible verification script using only an available runtime's standard library. Do not create a web UI, server, or use external data.