Manuscript IX-Free-2@1 · IX-Free-2.tex · theorem · abb2397a

Theorem: The required actual-space separations

Gate S: boundary-charge separation › An explicit sharp trace witness · lines 1170–1203 · proof included or linked

IX-Free-2@1#thm:separation

The intersections are

B \cap L_{2} = {0}, B \cap R_{2} = {0}, B \cap P_{mc} = {0} .

Consequently,

(L_{2} \otimes B) \cap (B \otimes R_{2}) = {0} .

Proof. Every element of $B$ is $B_{f}$ for one test $f$ , because the generating map is linear. A nonzero such element has $f \neq = 0$ . Lemma lem:sharpcharge gives a configuration at which its metric derivative has nonzero charge.

Every element of either partial-contraction space or the old algebra has zero charge at that configuration by Proposition prop:zerocharge. Equality of actual functionals would imply equality of their actual metric derivatives and intrinsic charges, a contradiction.

For algebraic tensor products,

(U \otimes V) \cap (U^{'} \otimes V^{'}) = (U \cap U^{'}) \otimes (V \cap V^{'}) .

Apply this identity to the displayed degree-two domains.

Displayed formulas in this passage

unlabeled · line 1173
$B \cap L_{2} = {0}, B \cap R_{2} = {0}, B \cap P_{mc} = {0} .$
unlabeled · line 1179
$(L_{2} \otimes B) \cap (B \otimes R_{2}) = {0} .$
unlabeled · line 1198
$(U \otimes V) \cap (U^{'} \otimes V^{'}) = (U \cap U^{'}) \otimes (V \cap V^{'}) .$

Theorem: The required actual-space separations

Displayed formulas in this passage

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