A Random Gain Buys No Seal: Stochastic Matching and Paper #54's Entry 9

Works paper #54's Entry 9 (item D-041-080-22, grade B) at the sharpened price of paper #522 Sec.6 (a)-(c), on the construction horn, by stating and exhausting a fifth class of pairing constructions -- the canonical double sealed by one stochastic (ensemble) matching at the scar, u_x(0, t) = LAM*u(0, t) at all times, where the gain LAM is drawn once from one stated distribution mu and held fixed thereafter. Gates reproduce paper #522's operator and fork exhibits, paper #535's priced grounds, paper #541's class-B exhibits, paper #553's class-N exhibits (72.032981 / 0.227324 / 1.539601 recomputed and asserted), paper #587's class-Q exhibits, and paper #599's class-R exhibits digit-for-digit first. The exhaustion is the realised-member lemma: for each drawn value the construction is paper #541's class-B member at that gain, whose exhibited field at k = 3 leaks 3|lam|/2, so every seal property of an ensemble is decided by the mass it puts on lam = 0. Exhibited shapes: the degenerate ensemble at gain 1 leaks deterministically at 1.500000; the symmetric ensemble on {-1, +1} has pointwise expected flux identically zero while every realised member leaks at 1.500000 (a seal in expectation, priced vacuous under clause (b)); the uniform ensemble on [-1, 1] and the standard normal ensemble leak almost surely, with expected largest fluxes 0.750000 and 1.196827; the mixed ensemble one-half point mass at 0 plus one-half at 2 seals with probability one half and leaks at 3.000000 with the other half. An ensemble seals almost surely exactly when mu is the point mass at zero, whose realised condition is class B's lam = 0 member restated a third time -- an unforced selection (the datum distinguishes 0 of the 6 sampled ensembles) admitting exactly the even parity family. The rival two-point-kernel candidate is printed as a collapse into class N (paper #553's second species restated) and was not worked as a class; the periodic-in-time species stands invited and unworked, printed as such. No member of class S meets clauses (a), (b) and (c) together. Disposition: strengthened, not closed; clauses (a)-(c) unchanged; a survivor must be a construction outside classes B, N, Q, R and S or a derivation down a priced route. Producing-count 0. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.