An Infinite-Harmonic Gain Buys No Seal: Absolutely Summable Periodic Matching and Paper #54's Entry 9

Works paper #54's Entry 9 at the sharpened price of paper #522 Sec.6 (a)-(c), on the construction horn, by stating and exhausting a seventh class of pairing constructions -- the canonical double sealed by one absolutely summable periodic matching at the scar, u_x(0, t) = LAM(t)*u(0, t) at all times, where LAM is one stated periodic coefficient whose Fourier coefficients are absolutely summable, the species proper having infinitely many nonzero harmonics. The class is the boundary species paper #626 printed as distinct and standing invited, unworked, together with its printed observation (quoted word for word in the manuscript) that a sealing infinite-harmonic coefficient must silence the flux factor at every mode above every finite bound. Gates reproduce the whole body of paper #626'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, paper #616's class-S exhibits, and paper #626's class-T exhibits (fluxes 1.500000 / 0.928502 / 1.926184 / 1.320129 / 5.608399, the vacuous sine seal). The exhaustion turns on the admission obstruction: the product of a single mode with the coefficient has constant coefficient c_k/2, nonzero at every mode the coefficient reaches, and the class's own correction -- a second mode at amplitude -c_1/c_2 -- cancels it exactly and admits a field whose flux is the coefficient times a nonzero trigonometric polynomial, not identically zero. Exhibited shapes: the geometric member leaks at 5.158728; the Poisson-kernel-shaped member at 4.099036; a member peaked at its fifth harmonic at 9.100471; a gapped member, whose lowest mode is admitted singly, at 0.109935; a sign-changing summable member at 7.548200; a coefficient with a sine part admits the zero field alone (the obstruction computed at 0.312500), a vacuous seal. The zero coefficient seals and is class B's lam = 0 member restated a fifth time -- an unforced selection (the datum distinguishes 0 of the 7 sampled coefficients) admitting exactly the even parity family. No member of class V 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, T and V or a derivation down a priced route; the incommensurate-period, field-driven and random periodic species stand untouched and still invited. 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.