Manuscript IX-E@1 · IX-E.tex · proposition · 8f8d5e7f

Proposition: First-stage commutator

E1: ordered-data consistency › The complete first-stage commutator calculation · lines 601–627 · proof included or linked

IX-E@1#prop:commutator, IX-E@1#eq:commutator

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On the declared first-stage sector,

B_{f} ⋆_{c} W_{g} - W_{g} ⋆_{c} B_{f} C_{f, g} = ℏ C_{f, g}, = - 2 ⟨ f (x) g (y) (K - K) (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩ = - 4 i ⟨ f (x) g (y) Im K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩ .

Proof. Subtract the two inherited first-stage free products. Their classical terms agree. Reorder only the smooth configuration factors and the scalar test factors in the second expression of (eq:locked-q). Distributional linearity gives (eq:commutator). Finally, $K - K = 2 i Im K$ .

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IX-E@1#eq:commutator · line 603
$B_{f} ⋆_{c} W_{g} - W_{g} ⋆_{c} B_{f} C_{f, g} = ℏ C_{f, g}, = - 2 ⟨ f (x) g (y) (K - K) (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩ = - 4 i ⟨ f (x) g (y) Im K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩ .$

Proposition: First-stage commutator

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