Manuscript XIII-B@1 · XIII-B.tex · proposition · b8a5c0f5

Proposition: Correct causal factorization

B2: the binary convention and the frozen extension class › The convention and its forced ordered values · lines 621–643 · proof included or linked

XIII-B@1#prop:factorization, XIII-B@1#eq:ordered-values

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Whenever the full functional supports of the entries are strictly time ordered, a binary time-ordered datum satisfying causal factorization must obey

supp V_{g} later than supp O : supp O later than supp V_{g} : T_{2}^{ren} = V_{g} ⋆_{H} O, T_{2}^{ren} = O ⋆_{H} V_{g}, R_{1} R_{1} = 0, = \frac{i}{ℏ} [O, V_{g}]_{⋆_{H}} .

The supports here are functional supports, not just $supp f$ .

Proof. Substitute the two time-ordered factorization values in (eq:R-definition). No extension on a collision diagonal can alter these values on the corresponding separated ordered region.

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XIII-B@1#eq:ordered-values · line 626
$supp V_{g} later than supp O : supp O later than supp V_{g} : T_{2}^{ren} = V_{g} ⋆_{H} O, T_{2}^{ren} = O ⋆_{H} V_{g}, R_{1} R_{1} = 0, = \frac{i}{ℏ} [O, V_{g}]_{⋆_{H}} .$

Proposition: Correct causal factorization

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