Step 1 – Michael addition: in 98% H₂SO₄ the β-keto ester (C-6, bearing CO₂Me) adds to methyl vinyl ketone, installing a 3-oxobutyl chain on C-6.
Step 2 – Aldol condensation: the methyl ketone of that chain condenses with C-2 (the α-bromo carbon, alpha to the ring C=O), closing a bridged cyclohexenone ring and making C-2 quaternary (C–Br, ring C=O, new C(Me)=C).
Step 3 – Quasi-Favorskii (semibenzilic) rearrangement: the lithiated dimethyl methylphosphonate adds to the ring ketone; the alkoxide pushes the C1–C6 bond onto C-2 and Br⁻ leaves. The six-membered ring contracts, C-6 becomes bonded to C-2, and C-1 leaves as the exocyclic C(=O)CH₂P(O)(OMe)₂. This gives the cis-fused bicyclo[3.3.0]octene shown.
Order: Michael addition → Aldol condensation → Quasi-Favorskii rearrangement. Answer: (c).