Manuscript IX-C-ST@1 · IX-C-ST.tex · proposition · 892f50c4

Proposition: Direct decomposition of the new sum

Whole-domain decomposition and the truncated product · lines 2169–2201 · proof included or linked

IX-C-ST@1#prop:newdirect, IX-C-ST@1#eq:newdirect

The six new domains form an algebraic direct sum:

D_{new} = (B \otimes W) \oplus (W \otimes B) \oplus (L_{4} \otimes B) \oplus (B \otimes R_{4}) \oplus (L_{2} \otimes B) \oplus (B \otimes R_{2}) .

Proof. Count metric and scalar slots in each argument. The argument-species bidegrees are respectively

((1, 1); (2, 0)), ((2, 0); (1, 1)), ((3, 1); (1, 1)), ((1, 1); (3, 1)),

((1, 1); (1, 1)), ((1, 1); (1, 1)) .

Polynomial homogeneity projects every relation onto these components. Only the last two share a component; their intersection is zero by Theorem thm:separation. Thus the entire sum, not merely its pairwise overlaps, is direct.

Displayed formulas in this passage

IX-C-ST@1#eq:newdirect · line 2172
$D_{new} = (B \otimes W) \oplus (W \otimes B) \oplus (L_{4} \otimes B) \oplus (B \otimes R_{4}) \oplus (L_{2} \otimes B) \oplus (B \otimes R_{2}) .$
unlabeled · line 2188
$((1, 1); (2, 0)), ((2, 0); (1, 1)), ((3, 1); (1, 1)), ((1, 1); (3, 1)),$
unlabeled · line 2192
$((1, 1); (1, 1)), ((1, 1); (1, 1)) .$

Proposition: Direct decomposition of the new sum

Displayed formulas in this passage

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