A Periodic Gain Buys No Seal: Floquet 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 sixth class of pairing constructions -- the canonical double sealed by one periodic-in-time (Floquet) matching at the scar, u_x(0, t) = LAM(t)*u(0, t) at all times, where LAM is one stated trigonometric polynomial of period 2 pi whose harmonic couplings read mode k off at modes k +/- n with weight c_n/2. The class is the species paper #616 printed as distinct and standing invited, unworked. Gates reproduce the whole body of paper #616's notebook digit-for-digit first: 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, paper #599's class-R exhibits, and paper #616's class-S exhibits. The exhaustion is the high-mode lemma: for a nonzero pure cosine coefficient with highest harmonic N, any mode k > N is admitted with its partner built from the Floquet couplings, and its flux (A^2 k / 2) sin(2kt) LAM(t) is a product of two trigonometric polynomials, not identically zero. Exhibited shapes: the constant member leaks at 1.500000 (class B restated); LAM = cos t leaks at 0.928502 through its k = 2 mode (its k = 1 mode is refused by its own constant term 0.500000); LAM = 1 + cos t leaks at 1.926184; LAM = cos 2t leaks at 1.320129; LAM = 2 cos t + cos 3t leaks at 5.608399; a coefficient with a sine part admits the zero field alone (the obstruction computed at 0.500000), a vacuous seal. The zero coefficient seals and is class B's lam = 0 member restated a fourth time -- an unforced selection (the datum distinguishes 0 of the 6 sampled coefficients) admitting exactly the even parity family. No member of class T 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, S and T or a derivation down a priced route. The sever-instant profile was not worked, derived, characterised or adopted; the identification route was not worked. Producing-count 0. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.