The Completion Class, Narrowed: One Cord Killed by Stability, the Action Proven Exactly Quadratic, and the Gathering's Two Debts Confirmed Against the Whole Corpus

My program's summit carries two missing objects. The first is the cord's constitutive law: the function g by which torsion does work in the neck's core, where the twist gradient reaches the Aldred bound and the certified quadratic corner is silent. The second is the measure density at the seam: a derived, not declared, road from the gathering's waking point f_W = 0.773950 to the residue 0.0275 and the toll 3Ω_Λ = 2.055. This paper attacks both with everything the corpus owns, and reports exactly what the math earned. On the cord: the rational completion g = y²/(1+y²) is killed — its energy is non-convex on an interval inside the core the tube must occupy, and its longitudinal wave speed goes imaginary on the same interval; one kill, counted once, at premise grade. The priced class narrows from three completions to two, { Born–Infeld, logarithmic }, and the phase-slip factor tightens to A_ps = g(1) ∈ [0.693147, 0.828427] — a spread of 1.195× where the record previously carried 1.66×. Behind the kill stands a structural result: the tube constitution's action, taken exactly as written, is polynomial of degree ≤ 2 in the torsion variables and induces g(y) = y² at every order. No saturating completion is a correction to that action; it must enter as a new term, a new field, or new matter-sector data — and inside the surviving class a single number, the y⁴ coefficient a_4 ∈ {−1/4, −1/2}, is all that separates the two remaining candidates. On the gathering: the wake point's priced failure is re-verified line by line and then extended from its original four sources to the whole corpus — twenty-six engine papers and six summit files — with zero hits: no arrival law, no volume law, no value of R(1) exists anywhere in the record. One error is found and corrected in the earlier paper's presentation (its miss range mixed two neighbour choices; §6). Inside the cap-geometry dress, a species result is proved: the boundary condition R(0) = 0 holds if and only if the arrival is late-weighted, T(f) = o(f²) — a species no premise in the corpus supplies. The sever is shown not to pin R(1); the counting door's missing object is named to the species (a grain-individuation map plus an involutive partner rule); and the mesoscopic-radius door, re-priced against the narrowed completion class, is closed for every surviving completion by the balance mechanism. The vacuum catastrophe is NOT SOLVED. The producing-count is 0 at derivation grade. What this paper earned is narrower and checkable: two intervals tightened, one completion buried, one theorem about where a completion can live, one species named for a missing law, and both debts re-priced against the entire record instead of the parts of it that were open on the desk.