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

Proposition: Constructed-block reversal

F8: reversal, presentation independence, and observations · lines 1362–1399 · proof included or linked

IX-Free@1#prop:reversal

The two-edge outputs satisfy

G_{12, 23} (f, g, f^{'})^{*} G_{12, 13} (f, g, f^{'})^{*} = G_{12, 23} (f ˉ^{'}, g ˉ, f ˉ), = G_{13, 23} (f ˉ^{'}, g ˉ, f ˉ) .

The constructed degree-two coefficient maps satisfy

L_{j} (F, B)^{*} = R_{j} (B^{*}, F^{*}), j = 0, 1, 2.

Proof. The free scalar kernel has

S_{μν} (x, x^{'}) = S_{ν μ} (x^{'}, x) .

Conjugation of a defined wavefront product is the product of the conjugated factors. Exchanging the endpoint variables therefore gives the two displayed graph identities.

The locked first-stage coefficients obey

Q_{f, g}^{*} = Q^{R}_{g ˉ, f ˉ} .

The graph identities give reversal at order one. At order two, the inherited parent symmetry and real $c$ give

N_{c} (f, g, f^{'}) = N_{c} (f ˉ^{'}, g ˉ, f ˉ) .

Order zero follows from classical conjugation.

Displayed formulas in this passage

unlabeled · line 1365
$G_{12, 23} (f, g, f^{'})^{*} G_{12, 13} (f, g, f^{'})^{*} = G_{12, 23} (f ˉ^{'}, g ˉ, f ˉ), = G_{13, 23} (f ˉ^{'}, g ˉ, f ˉ) .$
unlabeled · line 1372
$L_{j} (F, B)^{*} = R_{j} (B^{*}, F^{*}), j = 0, 1, 2.$
unlabeled · line 1379
$S_{μν} (x, x^{'}) = S_{ν μ} (x^{'}, x) .$
unlabeled · line 1388
$Q_{f, g}^{*} = Q^{R}_{g ˉ, f ˉ} .$
unlabeled · line 1394
$N_{c} (f, g, f^{'}) = N_{c} (f ˉ^{'}, g ˉ, f ˉ) .$

Proposition: Constructed-block reversal

Displayed formulas in this passage

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