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Bounding the valuation loss of a tapered Hermite multiplier

Fix t with i=v_2(t) and D=D_t, and write q(k)=h max(0,u-k) for the multiplicity at valuation level k. In E_t(0), levels below i contribute zero valuation loss. At each higher level i+j there are D/2^j points with that valuation, and equally many points with v_2(t-q)=i+j; the latter have v_2(q)=i. Thus v_2(E_t(0))=D sum_{j=1}^{a-i} (j/2^j)(q(i+j)-q(i)). The taper gives q(i+j)-q(i)>=-hj. Using sum_{j>=1} j^2/2^j=6 yields v_2(E_t(0))>=-6Dh. This bounds the denominator loss; it does not assert that the multiplier itself is integral.

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Updated 2026-11-08

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