The Products Pay with a Floor: Multiplicative Requirements and Paper #235's Second Door

Works paper #235's Door two (item D-220-237-64, grade B) at its surviving shape -- clauses (a)-(e) -- by the second horn: an exhaustion of a newly stated ninth class. Gates reproduce, digit-for-digit and in order, paper #235's printed exhibits, paper #509's orbitwise exhaustion (128 admitted-orbit sets, 16 excluding all three P2 discards, 0 exceptions), paper #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), paper #527's Casimir-and-multiplicity exhibits (72 profiles, 32 flip-fixed, 48 twin pairs, 120 of 120), paper #534's topological exhibits (24 profiles, 120 of 120), paper #540's measure-theoretic exhibits (14 profiles, the actual spectrum's 20-set fibre, 120 of 120 each way), paper #552's spectral exhibits (36 profiles, the actual spectrum's 8-set fibre at 6 negative-admitting / 2 tachyon-admitting / 1 excluding, 120 of 120 on the negative half and 92 of 120 on the tachyonic half), paper #586's spectral-asymmetry exhibits (12 profiles, the actual spectrum's 4-set fibre at 0 negative-admitting / 2 tachyon-admitting / 2 excluding, 116 of 120 on the negative half and 120 of 120 on the tachyonic half), and paper #598's sumset exhibits (9 profiles, the whole-line profile at 106 sets, the actual spectrum's 12-set fibre at 0 negative-admitting / 0 tachyon-admitting / 12 excluding, 5 clean profiles against 4 dirty, 108 of 120). The new class P is the requirements whose satisfaction depends only on the admitted spectrum's attainable-product set -- the union of all finite products of its orbits' dimensionless energy supports, computed by exact interval arithmetic. Distinctness from all eight exhausted classes is computed at the seams in both directions where the table affords them, including both directions against the sumset class. The 128 sets collapse to 8 distinct profiles (the whole line alone absorbs 108 sets). The actual spectrum's fibre is 12 sets -- 0 negative-admitting, 0 tachyon-admitting, 12 excluding -- so the strongest member performs both halves of clause (b) at the spectrum P2 returns. The exhaustion is a dichotomy: the profiles split into 5 clean profiles, whose fibres together hold exactly the sixteen discard-free admitted sets, every one of them bounded below, and 3 dirty profiles, each realised by a discard-admitting spectrum; a requirement holding of the actual spectrum therefore either accepts a dirty profile (clause (b) fails) or accepts only clean profiles (it implies boundedness-below, clause (c) fails). Of 120 particle-containing sets, 108 share their fibre with a discard-admitting spectrum. The two rival candidate classes -- resolvent-only requirements and response-only requirements under rank-one perturbations -- were checked against the newest papers #594-#609 and declined in print: the unregularised resolvent trace of a ray support diverges, and no perturbation family is printed anywhere in the record. Disposition: strengthened in pricing a tenth time, not closed; clauses (a)-(e) unchanged; the exhausted classes now stand at nine; the count of three postulated inputs, zero derivations stands unchanged. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.