The Topology Cannot See the Sheet: Topological Requirements and Paper #235's Second Door

Works paper #235's Door two (item D-220-237-64, grade B) at its Wave 20 surviving shape -- clauses (a)-(e) -- by the second horn: an exhaustion of a newly stated class. Gates reproduce, digit-for-digit and in order, paper #235's printed exhibits, packet #509's orbitwise exhaustion (128 admitted-orbit sets, 16 excluding all three P2 discards, 0 exceptions), packet #518's relational exhibits (pair condition 22 of 128 with 6 unbounded satisfiers and all 98 mixed sets violating; ordering condition 8 of 128, the kept spectrum not among them), and packet #527's Casimir-and-multiplicity exhibits (72 profiles, 32 flip-fixed, 48 twin pairs, 120 of 120, m-squared >= 0 at 64 sets with 48 of 64). The new class T is the requirements whose satisfaction depends only on the topological data of the admitted orbits' energy supports (closed ray, point, or whole real line, as abstract spaces). The reflection of the energy line is a homeomorphism, so the sheet-flip preserves every class-T profile: the 128 sets collapse to 24 distinct profiles, the spectrum P2 returns has a profile-identical twin admitting both negative-energy classes, and 120 of the 120 particle-containing admitted sets share their profile with a negative-admitting spectrum -- no class-T requirement excludes the negative-energy classes while holding of any particle spectrum. The class cannot even state boundedness-below, so clause (c) is passed for free and buys nothing. Its member 'no whole-line support' is satisfied by 64 sets and excludes the tachyonic orbit in all of them, yet admits a negative-energy class in 48 of the 64: class T performs exactly one half of P2's joint discard. Disposition: strengthened in pricing a fifth time, not closed; clauses (a)-(e) unchanged; the exhausted classes now stand at four; the count of three postulated inputs, zero derivations stands unchanged. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.