Manuscript IX-Free@1 · IX-Free.tex · proposition · 33b20e96

Proposition: Conditional completed-sector coherence

F7: four-order coherence, conditional on the missing maps · lines 1306–1339 · proof included or linked

IX-Free@1#prop:conditionalcoherence

Suppose the tests in Proposition prop:remainingdescent hold and the coefficient prescriptions glue consistently with their specified overlaps. With the same fixed real $c$ , the two associations then agree through order $ℏ^{3}$ . Their common coefficients are

ℏ -order 0123 common actual output A W C G_{12} + G_{13} + G_{23} G_{12, 23} + G_{12, 13} + G_{13, 23} N_{c} (f, g, f^{'}) 1.

Proof. Order zero is classical associativity. At the three positive orders, substitute (eq:Ldegree2)–(eq:Rdegree2scalar) and the descended versions of (eq:lambdaL)–(eq:lambdaR). This gives, respectively,

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}) .

Each is an equality of the displayed actual polynomial functionals. Equality of all four coordinates is equality in $V_{Σ}^{[3]}$ .

Displayed formulas in this passage

unlabeled · line 1329
$L_{1} (P_{L}, C) + Q_{L} C L_{2} (P_{L}, C) + L_{1} (Q_{L}, C) L_{2} (Q_{L}, C) = R_{1} (A, P_{R}) + A Q_{R}, = R_{2} (A, P_{R}) + R_{1} (A, Q_{R}), = R_{2} (A, Q_{R}) .$
unlabeled · line 1313
$ℏ -order 0123 common actual output A W C G_{12} + G_{13} + G_{23} G_{12, 23} + G_{12, 13} + G_{13, 23} N_{c} (f, g, f^{'}) 1.$

Proposition: Conditional completed-sector coherence

Displayed formulas in this passage

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