Manuscript II@2 · II.tex · proposition · 50d64def

Proposition: Complete response factorization

Ward reduction and coupling factorization › Factorization on the whole locked domain · lines 901–918 · proof included or linked

II@2#prop:factorization, II@2#eq:factorization

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On $X$ ,

r (d, θ; β) = q (d) m (β) + ℓ \sum θ_{ℓ} n_{ℓ} (β) .

Proof. The displayed species cases exhaust all four-leg words. They establish $F (d) = q (d) F (d_{*})$ when leg one is replaced. Simultaneous relabeling moves any replaced leg to position one. The additional quartic contacts contribute linearly by definition.

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II@2#eq:factorization · line 904
$r (d, θ; β) = q (d) m (β) + ℓ \sum θ_{ℓ} n_{ℓ} (β) .$

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section: Factorization on the whole locked domain
section: Factorization on the whole locked domain