The Power Sums Pay with a Floor: Fixed-Order Power-Sum Requirements and Paper #235's Second Door

Works paper #235's second door at clauses (a)-(e) word for word, on the second horn, by stating and exhausting a twelfth class of requirements -- class Z, requirements depending only on the admitted spectrum's power sum of one fixed stated order, the signed sum of E_i^p over the admitted orbits for one order p fixed in advance, where each orbit carries its definite energy where it has one (massive +1 / -1, vacuum 0) and the massless, continuous-spin and tachyonic orbits carry none, their energies sweeping [0, inf) or the whole line, so the datum is the pair (flag, finite sum) with flag D when any admitted orbit has no definite energy. Gates reproduce the whole body of paper #636's notebook digit-for-digit first (papers #235, #509, #518, #527, #534, #540, #552, #586, #598, #615, #627's candidate declinatures and class-X exhibits, and #636's with-repetition declinature and class-Y exhibits). The candidate was taken first in paper #636's printed order of untaken candidates (the ratio set stands second, not taken) and was tested, not assumed distinct: the orders collapse by parity (order 3's datum equals order 1's and order 4's equals order 2's on every set); the even datum is a function of the spectral class's profile -- P_2(S) = (D iff b_H(S) > 0, a_H(S)) on all 128 sets -- and is declined as a class (a coarsening of an exhausted class, its sharing direction against class H empty over all pairs); the threshold reading (each orbit's energy replaced by its support's finite endpoint) is set aside on statement grounds with its computed shape printed (6 profiles; its finite part is the spectral-asymmetry class's first coordinate on every set); the joint datum of all orders determines each fixed order over all pairs and is declined as a refinement on paper #627's standard. The odd datum is distinct in both directions from all eleven exhausted classes, by exhibited pairs against each and by exhaustive verification against the nine profile classes -- including both directions against the sumset and multiplicative closures, the complete-system class and the pairwise-sum class. The 128 admitted-orbit sets collapse to 6 profiles in each parity form; exactly 1 profile is clean in each form (odd (F, 1), realised by {massive positive} and {massive positive, vacuum}; even (F, 0), realised by the empty set and {vacuum}), its fibre's 2 sets both bounded below, and the other 5 are dirty. The actual spectrum's own profile (D, 1) is dirty in both parity forms -- odd fibre 30 sets (16 negative-admitting, 16 tachyon-admitting, 6 excluding), even fibre 60 sets (46 negative-admitting, 32 tachyon-admitting, 6 excluding) -- so every class-Z requirement holding of the actual spectrum fails clause (b) at the spectrum itself, and one accepting only clean profiles fails clause (c): the class performs clause (b) at the actual spectrum not at all, a failure shape no predecessor exhibited. 118 of 120 (odd) and 120 of 120 (even) particle-containing sets share a fibre with a discard-admitting spectrum. Disposition: strengthened in pricing a thirteenth time, not closed; clauses (a)-(e) unchanged; the exhausted classes now stand at twelve. Count unchanged: three postulated inputs, zero derivations. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.