Manuscript VI@1 · VI.tex · proposition · b5b60afb

Proposition: Conditional coefficient identity

Second-order retarded bookkeeping and the missing quantum step › The algebraic four-term identity · lines 1863–1889 · proof included or linked

VI@1#prop:four-terms, VI@1#eq:four-terms

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If the retarded products in (eq:retarded-series) exist on the listed inputs and have the stated multilinearity, then

[R_{V} (F)]_{κ^{2}} = O_{f}^{(2)} + R_{1} (V_{1}; O_{f}^{(1)}) + R_{1} (V_{2}; O_{f}^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; O_{f}^{(0)}) .

Proof. The zeroth retarded term contributes $O_{f}^{(2)}$ . The first retarded term contributes

R_{1} (V_{1}; O_{f}^{(1)}) + R_{1} (V_{2}; O_{f}^{(0)}) .

The second retarded term contributes $\frac{1}{2} R_{2} (V_{1}, V_{1}; O_{f}^{(0)})$ . Terms with three or more copies of $V$ have order at least $κ^{3}$ .

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VI@1#eq:four-terms · line 1867
$[R_{V} (F)]_{κ^{2}} = O_{f}^{(2)} + R_{1} (V_{1}; O_{f}^{(1)}) + R_{1} (V_{2}; O_{f}^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; O_{f}^{(0)}) .$
unlabeled · line 1882
$R_{1} (V_{1}; O_{f}^{(1)}) + R_{1} (V_{2}; O_{f}^{(0)}) .$

Proposition: Conditional coefficient identity

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