λEnter Calculus CompanionRegistered manuscripts, kept distinct

Manuscript VI@1 · VI.tex · figure · b5b60afb

Exact first-order switch defect

An exact compact-switch obstruction › The candidate and the tested identification · lines 1490–1507

VI@1#fig:VI-switch-defect

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
(h,ϕ0)↦(h+Kξ,ϕ0)(h,\phi_0)\mapsto(h+K\xi,\phi_0)
coordinate branchδw=ξ\delta w=\xiδ(−Xwϕ0)=−Xξϕ0\delta(-X_w\phi_0)=-X_\xi\phi_0
switched-response branchδϕ1,g=gXξϕ0+GR[(∂μg)Kξ‾μν∂νϕ0−[P,g]Xξϕ0]\delta\phi_{1,g}=gX_\xi\phi_0+G_{\mathrm R}\bigl[(\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi_0-[P,g]X_\xi\phi_0\bigr]
δAf,g(1)=∫f[(g−1)Xξϕ0+GR(switching terms)]\delta\mathcal A_{f,g}^{(1)}=\int f\bigl[(g-1)X_\xi\phi_0+G_{\mathrm R}(\text{switching terms})\bigr][P,g]u=(□ηg)u+2(∂μg)∂μu[P,g]u=(\Box_\eta g)u+2(\partial^\mu g)\partial_\mu u

Exact classical retarded identity. Not a loop anomaly or a quantum Master Ward Identity.

Exact first-order switch defect. The geometric compensator changes by , while the compactly switched scalar response supplies and additional retarded switching terms. Their difference gives the displayed classical defect. A nonzero defect here is not a computed quantum anomaly.

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