Manuscript IX-C-ST@1 · IX-C-ST.tex · theorem · 892f50c4

Theorem: Coefficientwise coherence criterion

Descent, scalar tests, and coherence equations · lines 1309–1331 · proof included or linked

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

Assuming that the displayed second-stage maps exist on actual arguments and that the scalar coefficient is supplied only by the degree-two block, the two associations agree modulo $ℏ^{4}$ exactly when

L_{1} (P, B_{f^{'}}) + Q B_{f^{'}} L_{2} (P, B_{f^{'}}) + L_{1} (Q, B_{f^{'}}) L_{2, c} (Q, B_{f^{'}}) = R_{1} (B_{f}, P^{'}) + B_{f} Q^{'}, = R_{2} (B_{f}, P^{'}) + R_{1} (B_{f}, Q^{'}), = R_{2, c} (B_{f}, Q^{'}) .

Proof. Expand the first stage $P + ℏ Q$ and then its second multiplication, and do the same for $P^{'} + ℏ Q^{'}$ . The order-zero equality is classical associativity. Equating the next three coefficients gives exactly (eq:assoc1)–(eq:assoc3).

Displayed formulas in this passage

IX-C-ST@1#eq:assoc1 · line 1315
$L_{1} (P, B_{f^{'}}) + Q B_{f^{'}} L_{2} (P, B_{f^{'}}) + L_{1} (Q, B_{f^{'}}) L_{2, c} (Q, B_{f^{'}}) = R_{1} (B_{f}, P^{'}) + B_{f} Q^{'}, = R_{2} (B_{f}, P^{'}) + R_{1} (B_{f}, Q^{'}), = R_{2, c} (B_{f}, Q^{'}) .$
IX-C-ST@1#eq:assoc2 · line 1315
$L_{1} (P, B_{f^{'}}) + Q B_{f^{'}} L_{2} (P, B_{f^{'}}) + L_{1} (Q, B_{f^{'}}) L_{2, c} (Q, B_{f^{'}}) = R_{1} (B_{f}, P^{'}) + B_{f} Q^{'}, = R_{2} (B_{f}, P^{'}) + R_{1} (B_{f}, Q^{'}), = R_{2, c} (B_{f}, Q^{'}) .$
IX-C-ST@1#eq:assoc3 · line 1315
$L_{1} (P, B_{f^{'}}) + Q B_{f^{'}} L_{2} (P, B_{f^{'}}) + L_{1} (Q, B_{f^{'}}) L_{2, c} (Q, B_{f^{'}}) = R_{1} (B_{f}, P^{'}) + B_{f} Q^{'}, = R_{2} (B_{f}, P^{'}) + R_{1} (B_{f}, Q^{'}), = R_{2, c} (B_{f}, Q^{'}) .$

Theorem: Coefficientwise coherence criterion

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