No Derivation of S1: The Countermodel That Closes von Neumann's Front Door

Discharges, by impossibility, the first kill condition of house paper #161 (debt D-161-201-03): 'Derive S1 -- show that any completion reproducing the quantum statistics on average must have dispersion-free subensembles whose assignments are additive for non-commuting sums.' For two spin components at any angle phi strictly between 0 and 180 degrees, in the state |+n>, an explicit distribution over eigenvalue-respecting assignments (vA = +1 almost surely; vB = +/-1 with P(+) = (1+cos phi)/2; v(A+B) = +/-2cos(phi/2) with P(+) = (1+cos(phi/2))/2) reproduces the three quantum expectations exactly -- max residual 4.4e-16 across nine angles -- while the probability that any member satisfies additivity S1 is exactly zero, since {0, 2} and {+/-2cos(phi/2)} are disjoint. Average additivity plus the eigenvalue rule therefore cannot imply subensemble additivity; the requested derivation does not exist. Full-theory scope is carried by Bohm's 1952 model, an existing completion reproducing all quantum statistics whose assignments decline S1. The one sufficient extra premise, noncontextual functional composition (FUNC), is jointly unsatisfiable with the eigenvalue rule on the same exhibit by #161's own count (0 of 4). S1 is underivable in both directions. #161 loses its front door (the kill condition is closed, permanently) and keeps everything else: the conditional 1932 proof stands, the transmission debt and kill conditions 2-3 stand open, and nothing here touches quantum gravity or the vacuum ledger.