The Heat Cannot See the Sheet: Spectral Requirements and Paper #235's Second Door

Works paper #235's Door two (item D-220-237-64, grade B) at its Wave 22 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), 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), packet #534's topological exhibits (24 profiles, 120 of 120, no-whole-line member at 64 sets with 48 of 64 and 16 excluding), and packet #540's measure-theoretic exhibits (14 profiles, the actual spectrum's 20-set fibre at 16 negative-admitting / 10 tachyon-admitting / 1 excluding, 120 of 120 each way, the finite-measure member at 2 sets and 0 particle spectra). The new class H is the requirements whose satisfaction depends only on the admitted spectrum's heat trace -- equivalently its spectral-zeta values, equivalently its spectral-action coefficients. The integrand exp(-t E^2) is even, so the sheet pairs carry identical trace functions and the sheet-flip preserves every H-profile: the 128 sets collapse to 36 distinct profiles (counts a in 0..2, b in 0..5, c in 0..1, recovered from the trace exactly). Every particle-containing set shares its fibre with a negative-admitting spectrum (120 of 120); the tachyon half fails at 92 of 120 and, decisively, at the spectrum P2 returns, whose 8-set fibre holds 6 negative-admitting sets, 2 tachyon-admitting sets and 1 excluding set (itself). The class's tachyon-excluding member 'b <= 1' holds of 32 sets, 28 particle-containing, but not of the actual spectrum (b = 2). The strongest joint member, 'the total trace is the vacuum constant alone', satisfies 2 sets and no particle spectrum. The class cannot state boundedness-below, so clause (c) is passed for free and buys nothing. Disposition: strengthened in pricing a seventh time, not closed; clauses (a)-(e) unchanged; the exhausted classes now stand at six; the count of three postulated inputs, zero derivations stands unchanged. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.