Nonlinear Matching Buys the Seal with a Parity: A Third Stated Class of Pairing Constructions for 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 third class of pairing constructions -- the nonlinear class paper #553's scope note named as outside its class N. Gates reproduce, digit-for-digit and in order, both recovered versions of paper #54's Entry 9 price, paper #522's operator and fork exhibits (P^2 = I; handedness flip +0.5 to -0.5; eigenfield seal 0.000000; mixed-parity leak 4.500000; pulse flux 294.3; 24 pairings), paper #535's priced grounds (regularity 9.000000 / 0.000000 / 9.000000; finite energies 4.709563 / 4.290482 / 9.000045; Laplacian eigenvalues 0 / -2.467401 / -9.869604 / -22.206610), paper #541's class-B exhibits (leak 3|lambda|/2 at k = 3 with samples 0.375 / 0.75 / 1.5 / 3 / 6; sealing set exactly {lambda = 0, Dirichlet}; the datum distinguishes 0 members), and paper #553's class-N exhibits (largest scar fluxes 72.032981 and 0.227324; the cross-frequency leak 1.539601; the datum distinguishes 0 sampled members). The new class Q is paper #522's canonical double plus one pointwise nonlinear matching at the scar, u_x(0,t) = F(u(0,t)) at all times, for one fixed odd function F bounded on bounded intervals; its linear members are exactly class B's Robin family. Exhaustion by one construction: for any odd F and any amplitude A the scar pair w = A cos t, z = F(A cos t) is antiperiodic in its second entry, hence is the scar pair of an admitted finite-energy field, whose flux A sin t F(A cos t) vanishes identically only if F is zero on the whole range. Every nonzero F therefore leaks (exhibited: a member with slope at the origin at 0.375000; the cubic member, whose leak is generated at a harmonic its scar value did not carry, at 0.324760; a dead-zone member flat at the origin, sealed on its small even sector and leaking at amplitude 2 at 0.738017). The one sealing member, F identically zero, is paper #541's lambda = 0 member of class B restated -- a selection the datum distinguishes 0 of, admitting exactly the even parity family. No member of class Q meets (a)-(c) together. Disposition: strengthened, not closed; a survivor must come from outside classes B, N and Q or down one of paper #535's priced derivation routes. The sever-instant profile was not worked, derived, characterised, or adopted; the identification route was not worked; producing-count 0, printed. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.