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Lemma 8 ([The additive comparison for the $v_4$ targets]). Let s=qk+1 , w=u_4+k , k≥1 . A nonzero normalized cobar group in degree (s,w) with coefficients R/I_4 requires (q-1)k≤u_4-1, w≤(qu_4-1)/(q-1)<2u_4<p^4<u_5. For every such k , …

Lemma 8 ([The additive comparison for the targets]). Let s=qk+1 , w=u4+k , k≥1 . A nonzero normalized cobar group in degree (s,w) with coefficients R/I4 requires (q-1)k≤u4-1, w≤qu4-1q-1<2u4<p4<u5. For every such k , additive specialization is an isomorphism on cohomology in degree (s,w) , by an identification of the cochain groups of lengths s-1,s,s+1 . No v4 coefficient or t4 bar occurs in any of those three cochain groups.

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证明
Lemma 1 ([Leading coefficients and low cooperations]). For each m≥1 , I_m is an…Lemma 4 ([A comparison in three cochain degrees]). Suppose s≥1 , w<u_ m+1 , and…观点
The first two inequalities are the rearrangement of w≥s . Here u_4=1 , 028 , 262 , 820, 2u_4=2 , 056 , 525 , 640<p^4=1 , 036 , 488 , 922 , 561, so the remaining inequalities in Equation hold at the fixed prime. For a cochain length d∈s-1,s,s+1 , a coefficient containing v_4 would leave at most k weight for d≥qk>k reduced factors. It therefore cannot occur. A bar containing t_4 would leave at most k weight for the other d-1≥qk-1>k bars, and likewise cannot occur. Higher generators exceed the weight bound. The remaining coproducts are additive by Lemma 1 ([Leading coefficients and low cooperations]); equivalently, apply Lemma 4 ([A comparison in three cochain degrees]). This identifies both adjacent differentials and proves the cohomology assertion.
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