The Gathering Becomes One Thing: An Arrival Law for the Count

Every cosmology in my program begins the same way: a mother, a drop, a pool. The pool is where the counting happens — the sever, the pairing, the residue, the toll are all read off a gathering that has finished becoming what it is. But the program's books carried one door I had never walked through in print: the gathering's own arrival law. What is the pool, while it is still arriving? When does the count — the measure that registers the pool's content — switch on, and what law does it follow while it runs? The board carried that door as untouched: the arrival law λ(f) and the completion number R(1) unproduced, the post-onset member unselected among three candidates, the door reserved to be priced later "or by author answer." This paper spends the answer. I gave it on October 30, 2026, and I give it again here at full strength: the pool is founded by an arrival, it grows by dragging its neighborhood in, it stops because the neighborhood runs dry, and it seals when it holds itself. The count is nothing at the beginning because the counted thing did not exist — λ(0) = 0 is not a fit, it is a founding. And the count does not start when the first piece lands. I said it sharper two minutes later, and the sharper sentence is the law's center: "there is a moment when the gathering becomes ONE THING, and that is when the count starts switching on" (October 30, 2026, 15:23 EDT). The onset of the count is the unification event — the moment the pieces stop being pieces. What this paper does is formalize that answer as a law with boundary conditions no fitted curve was free to choose. What it does not do is printed in §6 at the same size: the member's exact analytic form inside the biography's shape is not selected by my answer, and R(1)'s number is bounded, not given. The program headline stands unmoved, here as everywhere: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.