Manuscript IX-D-ST@1 · IX-D-ST.tex · theorem · b1dc631a

Theorem: Conditional unnormalized causal subtraction

Conditional source contacts, subtraction, and order audit · lines 958–999 · proof included or linked

IX-D-ST@1#thm:conditionalret, IX-D-ST@1#eq:Rhat

Under Assumption ass:future,

Ret_{1} (d) = i (T_{2} (d) - S (d)), d \in E_{2},

has admitted coefficients and descendants, zero classical coefficient, and exact supplied old unnormalized overlaps. It satisfies

Ret_{1} (V, O) = 0 if supp_{fun} O \cap J_{ch}^{+} (supp_{fun} V) = \emptyset,

and

Ret_{1} (d)^{*} = Ret_{1} (γ d) .

A common named fixed-support continuity property passes to this difference.

Proof. Admission, descendant admission, and old overlap follow by linear subtraction. The order-zero coefficients cancel.

The support condition is $V ≳ O$ , so factorization makes the two terms equal. For involution use free reversal and (eq:antitime):

Ret_{1} (d)^{*} = - i (T_{2} (d)^{*} - S (d)^{*}) = - i (S (γ d) + S (σ γ d) - T_{2} (γ d) - S (σ γ d)) = i (T_{2} (γ d) - S (γ d)) .

Continuity follows from continuity of addition and scalar multiplication in the named locally convex range.

Displayed formulas in this passage

IX-D-ST@1#eq:Rhat · line 961
$Ret_{1} (d) = i (T_{2} (d) - S (d)), d \in E_{2},$
unlabeled · line 968
$Ret_{1} (V, O) = 0 if supp_{fun} O \cap J_{ch}^{+} (supp_{fun} V) = \emptyset,$
unlabeled · line 974
$Ret_{1} (d)^{*} = Ret_{1} (γ d) .$
unlabeled · line 988
$Ret_{1} (d)^{*} = - i (T_{2} (d)^{*} - S (d)^{*}) = - i (S (γ d) + S (σ γ d) - T_{2} (γ d) - S (σ γ d)) = i (T_{2} (γ d) - S (γ d)) .$