Position 2 racks in the upper and lower thirds of the oven and preheat to 400 degrees F.
Combine the tomatoes, capers, 3 tablespoons of the olive oil, lemon zest and juice, shallot, basil, 1/2 teaspoon salt, and a few grinds of black pepper in a medium bowl. Set aside to let the flavors develop.
Cut four 16-inch long sheets of parchment paper. Fold each sheet in half and open it like a book. Divide the pieces of parchment between 2 baking sheets (2 pieces per sheet).
Toss the corn and green beans with the remaining tablespoon of olive oil, 1/2 teaspoon salt, and a few grinds of black pepper in a bowl. Divide between the 4 parchment sheets, mounding the mixture on one side of the fold. Sprinkle each cod fillet with 1/4 teaspoon salt and place on top of a vegetable mound. Dot with butter and drizzle with wine.
Fold over the top half of the parchment, crimping and overlapping small folds along the open edge to seal tightly. Bake until the parchment puffs, about 15 minutes; the fish will be tender and cooked through. Transfer the packs to plates. Avoiding the hot steam, carefully cut open the pack. Spoon the reserved tomato-basil salsa with juices over each fillet and serve immediately.
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