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

Theorem: Coefficientwise associativity criterion

Second-stage coefficient equations and associativity · lines 1064–1090 · proof included or linked

IX-C@1#thm:coherence, IX-C@1#eq:assoc1, IX-C@1#eq:assoc2, IX-C@1#eq:assoc3

Assuming the maps in (eq:AL)–(eq:AR) are well defined, the equality $A_{L} = A_{R} mod ℏ^{4}$ holds if and only if the following three identities hold for every permitted $f, g, f^{'}$ :

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

The order-zero identity holds by classical associativity.

Proof. Equality in the truncated module is equality of its four coefficient coordinates. The zero-order terms coincide because $P_{L} C = (A W) C = A (W C) = A P_{R}$ . The other three coordinates are exactly (eq:assoc1)–(eq:assoc3).

Displayed formulas in this passage

IX-C@1#eq:assoc1 · line 1071
$L_{1} (P_{L}, C) + Q_{L} C L_{2} (P_{L}, C) + L_{1} (Q_{L}, C) L_{2} (Q_{L}, C) = R_{1} (A, P_{R}) + A Q_{R}, = R_{2} (A, P_{R}) + R_{1} (A, Q_{R}), = R_{2} (A, Q_{R}) .$
IX-C@1#eq:assoc2 · line 1071
$L_{1} (P_{L}, C) + Q_{L} C L_{2} (P_{L}, C) + L_{1} (Q_{L}, C) L_{2} (Q_{L}, C) = R_{1} (A, P_{R}) + A Q_{R}, = R_{2} (A, P_{R}) + R_{1} (A, Q_{R}), = R_{2} (A, Q_{R}) .$
IX-C@1#eq:assoc3 · line 1071
$L_{1} (P_{L}, C) + Q_{L} C L_{2} (P_{L}, C) + L_{1} (Q_{L}, C) L_{2} (Q_{L}, C) = R_{1} (A, P_{R}) + A Q_{R}, = R_{2} (A, P_{R}) + R_{1} (A, Q_{R}), = R_{2} (A, Q_{R}) .$

Theorem: Coefficientwise associativity criterion

Displayed formulas in this passage

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