The Shore: A Bounded Pool of Dark Energy, Stated as Geometry, and the Toll It Does Not Price

The water around the pair is the pool of dark energy, and the shore exists because of the one weighted ball in the middle. That is my statement, premise P-14, given October 29, 2026, at 12:58 EDT and formalized the same day. This paper spends the fourth part of that statement, S4, at the address it points to: cosmology. Stated as geometry, the claim is short. The stage axis has a physical edge. The pool of dark energy is bounded. Its boundary, the shore, is set by the central weighted pair, the mother and the baby read as one bound object, so the waterline is a function of the mass and content in the middle. Full gathering, f = 1, is gathering against that pool and no other. An observer inside the pool does not see the shore as an object in the sky. He measures the pool from inside: its present fraction, Omega_Lambda,0 = 0.685, and its equation of state, w(z), redshift by redshift. I register the toll before I price it. If this geometry prices any quantity the DESI fork can grade, it has to price one in the (w0, wa) plane: it has to select Branch T, Branch C, or a point of Branch D, or produce a new definite point or track of its own. Section 5 prints that registration in full, before any pricing runs. The toll comes back empty. A bounded pool with a constant inside, a bounded pool with a thawing inside, and a bounded pool with a freezing inside all satisfy every clause of the shore geometry, because the geometry states a boundary and says nothing about the pressure law inside it. Three interiors that exclude each other share one shore. So the geometry implies no (w0, wa), no sign for wa, and no w(z) track. Against every branch of the DESI fork, and against the two galaxy reductions filed beside that fork in the house record, the shore's grade is the same word: SILENT. This paper therefore converts, by its own registered rule, to what it honestly is: a stated geometry with an obstruction result attached. The shore denominates the axis. It prices nothing yet. The premise grade stays on the face of it. S4 is my statement, formalized, and its yield in the premise record is constraint grade only: it moves no number in the campaign that filed it. Nothing here raises any grade the frame already carries. The vacuum conversion is NOT SOLVED, the standing state of the wider program is PARTIAL, and this paper touches neither headline except to print it. Grades printed in the paper: Toll outcome: NO PRICED QUANTITY; against every branch of the DESI fork, and against the two galaxy reductions filed beside that fork, the shore's grade is SILENT (Branch T SILENT; Branch C SILENT; Branch D SILENT at each point; the fork itself SILENT); the vacuum conversion is NOT SOLVED and the standing state of the wider program is PARTIAL.