Manuscript IX-Free-2@1 · IX-Free-2.tex · theorem · abb2397a

Theorem: Wordwise coherence through order three

Gate C: coherence for the selected word · lines 1445–1500 · proof included or linked

IX-Free-2@1#thm:coherence

For every fixed real common parent $c$ ,

(A ⋆_{c} W) ⋆_{c} C = A ⋆_{c} (W ⋆_{c} C) in P_{Σ} \otimes_{C} C [ℏ] / (ℏ^{4}) .

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. The left association is

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),

and the right association is

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}) .

There is no $C_{3}$ contribution on these new blocks by the selected coefficient assignment.

Classical associativity gives equality at order zero. The descended degree-four prescriptions and inherited degree-two maps give

L_{1} (P_{L}, C) + Q_{L} C R_{1} (A, P_{R}) + A Q_{R} L_{2} (P_{L}, C) + L_{1} (Q_{L}, C) R_{2} (A, P_{R}) + R_{1} (A, Q_{R}) L_{2} (Q_{L}, C) = (G_{13} + G_{23}) + G_{12}, = (G_{12} + G_{13}) + G_{23}, = G_{13, 23} + G_{12, 23} + G_{12, 13}, = G_{12, 13} + G_{12, 23} + G_{13, 23}, = R_{2} (A, Q_{R}) = N_{c} (f, g, f^{'}) 1.

These are now equalities of coefficients of existing actual partial products. Equality of all four coefficients is equality in $P_{Σ} \otimes_{C} C [ℏ] / (ℏ^{4})$ .

Displayed formulas in this passage

unlabeled · line 1467
$P_{L} C + ℏ (L_{1} (P_{L}, C) + Q_{L} C) + ℏ^{2} (L_{2} (P_{L}, C) + L_{1} (Q_{L}, C)) + ℏ^{3} L_{2} (Q_{L}, C),$
unlabeled · line 1474
$A P_{R} + ℏ (R_{1} (A, P_{R}) + A Q_{R}) + ℏ^{2} (R_{2} (A, P_{R}) + R_{1} (A, Q_{R})) + ℏ^{3} R_{2} (A, Q_{R}) .$
unlabeled · line 1486
$L_{1} (P_{L}, C) + Q_{L} C R_{1} (A, P_{R}) + A Q_{R} L_{2} (P_{L}, C) + L_{1} (Q_{L}, C) R_{2} (A, P_{R}) + R_{1} (A, Q_{R}) L_{2} (Q_{L}, C) = (G_{13} + G_{23}) + G_{12}, = (G_{12} + G_{13}) + G_{23}, = G_{13, 23} + G_{12, 23} + G_{12, 13}, = G_{12, 13} + G_{12, 23} + G_{13, 23}, = R_{2} (A, Q_{R}) = N_{c} (f, g, f^{'}) 1.$
unlabeled · line 1448
$(A ⋆_{c} W) ⋆_{c} C = A ⋆_{c} (W ⋆_{c} C) in P_{Σ} \otimes_{C} C [ℏ] / (ℏ^{4}) .$
unlabeled · line 1454
$ℏ -order 0123 common actual output A W C G_{12} + G_{13} + G_{23} G_{12, 23} + G_{12, 13} + G_{13, 23} N_{c} (f, g, f^{'}) 1.$

Theorem: Wordwise coherence through order three

Displayed formulas in this passage

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