Manuscript IX-Free@1 · IX-Free.tex · proposition · 33b20e96

Proposition: Exact remaining degree-four descent tests

F5–F6: actual maps and the remaining overlap gates › Uncompleted degree-four blocks · lines 1186–1210 · proof included or linked

IX-Free@1#prop:remainingdescent

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The prescriptions (eq:lambdaL) define maps on $L_{4} \times B$ exactly when

λ^{L}_{j} ((ker p_{L}) \otimes T) = 0, j = 1, 2.

The right prescriptions define maps on $B \times R_{4}$ exactly when

λ^{R}_{j} (T \otimes (ker p_{R})) = 0, j = 1, 2.

Here the relation kernels are precisely the distributional equations in Proposition prop:prelations.

Proof. Necessity follows by evaluating a proposed map at a zero actual argument. For sufficiency, choose a presentation of the actual intermediate argument. Two choices differ by the relevant relation kernel, which the hypothesis annihilates. Bilinearity and uniqueness then follow.

Displayed formulas in this passage

unlabeled · line 1190
$λ^{L}_{j} ((ker p_{L}) \otimes T) = 0, j = 1, 2.$
unlabeled · line 1196
$λ^{R}_{j} (T \otimes (ker p_{R})) = 0, j = 1, 2.$

Proposition: Exact remaining degree-four descent tests

Displayed formulas in this passage

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