For the Gel of the Sea:
In a large pot, boil enough water to cover the littleneck clams. Add the clams to the pot, and cook until the clams open. Remove clams from the pot, and reserve for another use. Strain the water used for boiling the clams and reserve.
Meanwhile, in a large pot, boil enough water to cover the mussels. Add the mussels to the pot, and cook until they open. Remove the mussels, and reserve for another use. Strain the water used for boiling the mussels and reserve. Separately reduce the clam and mussel broths until they give off a strong sea aroma. Strain both broths separately through cheesecloth.
Blanch the Gillardeau oysters for 30 seconds in just enough water to cover them and immediately transfer them to an ice bath. Shuck the oysters and reserve the oyster liquor, cooking water, and meat separately. Reserve the shells.
Combine the clam and mussel broths with the water used to cook the oysters and the oyster liquor. Texturize the mixture with the thickening agent, taking care to maintain the correct proportion of thickening agent to liquid, according to the instructions provided by the brand of thickening agent. Blend the mixture thoroughly in a food processor. Add the mixture to a vacuum-sealed pouch and carefully remove all air bubbles.
For the Seawater Distillate:
Combine the seawater and the seaweed in a beaker. Put the beaker in an ultrasonic distillation machine for 40 minutes set on cycle 7 at 80 percent power. Strain and reserve.
To Assemble and Serve:
When ready to serve, add 20 grams of Gel of the Sea to the 4 reserved oyster shells. Add the meat of 1 oyster to each of the oyster shells. Arrange the Salicornia on top of each oyster. In a glass dish, add the sea pebbles and arrange the dry ice on top. Carefully arrange the oysters on top of the dry ice. Add the Seawater Distillate to a separate jar, which will be poured tableside over the oysters, creating a sea “mist” full of natural sea aromas.