Manuscript IX-C@1 · IX-C.tex · proposition · 03ce8e6b

Proposition: Formal coefficient expansion

Second-stage coefficient equations and associativity · lines 1033–1062 · proof included or linked

IX-C@1#prop:expansion, IX-C@1#eq:AL, IX-C@1#eq:AR

Under the selected contraction-degree prescription, the two candidate compositions expand as

A_{L} = A_{R} = P_{L} C + ℏ (L_{1} (P_{L}, C) + Q_{L} C) + ℏ^{2} (L_{2} (P_{L}, C) + L_{1} (Q_{L}, C)) + ℏ^{3} L_{2} (Q_{L}, C), A P_{R} + ℏ (R_{1} (A, P_{R}) + A Q_{R}) + ℏ^{2} (R_{2} (A, P_{R}) + R_{1} (A, Q_{R})) + ℏ^{3} R_{2} (A, Q_{R}) .

Their selected field degrees are $6, 4, 2, 0$ , respectively.

Proof. Expand the coefficientwise formal product of $P_{L} + ℏ Q_{L}$ with $C$ , and of $A$ with $P_{R} + ℏ Q_{R}$ . Collect equal powers of $ℏ$ . The degree-two sharp factor permits at most two cross-contractions. Every contraction removes two field slots, giving the degree ledger.

Displayed formulas in this passage

IX-C@1#eq:AL · line 1037
$A_{L} = A_{R} = P_{L} C + ℏ (L_{1} (P_{L}, C) + Q_{L} C) + ℏ^{2} (L_{2} (P_{L}, C) + L_{1} (Q_{L}, C)) + ℏ^{3} L_{2} (Q_{L}, C), A P_{R} + ℏ (R_{1} (A, P_{R}) + A Q_{R}) + ℏ^{2} (R_{2} (A, P_{R}) + R_{1} (A, Q_{R})) + ℏ^{3} R_{2} (A, Q_{R}) .$
IX-C@1#eq:AR · line 1037
$A_{L} = A_{R} = P_{L} C + ℏ (L_{1} (P_{L}, C) + Q_{L} C) + ℏ^{2} (L_{2} (P_{L}, C) + L_{1} (Q_{L}, C)) + ℏ^{3} L_{2} (Q_{L}, C), A P_{R} + ℏ (R_{1} (A, P_{R}) + A Q_{R}) + ℏ^{2} (R_{2} (A, P_{R}) + R_{1} (A, Q_{R})) + ℏ^{3} R_{2} (A, Q_{R}) .$

Proposition: Formal coefficient expansion

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