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

Compatible conclusions and order obstruction

An exact compact-switch obstruction › A second-order term cannot cancel the first-order gap · lines 1680–1699

VI@1#fig:VI-compatible-order-obstruction

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
geometric solution-jet actionδϕ1=Xξϕ0,δw=ξ\delta\phi_1=X_\xi\phi_0,\qquad \delta w=\xiδ(ϕ1−Xwϕ0)=0\delta(\phi_1-X_w\phi_0)=0
plateau switched assignmentδϕ1,g=0,δw=ξ\delta\phi_{1,g}=0,\qquad \delta w=\xiδ(ϕ1,g−Xwϕ0)=−Xξϕ0\delta(\phi_{1,g}-X_w\phi_0)=-X_\xi\phi_0
δA=κΔ1+κ2Δ2 mod κ3,Δ1≠0\delta\mathcal A=\kappa\Delta_1+\kappa^2\Delta_2\ \mathrm{mod}\,\kappa^3,\qquad \Delta_1\neq 0
Δ2\Delta_2 cancels Δ1\Delta_1

Formal equality is coefficientwise. A positive ℏ\hbar-order cannot cancel a fixed ℏ0\hbar^0 coefficient.

Compatible conclusions and order obstruction. Classical descent uses the full scalar coefficient action, which cancels the coordinate variation. The specified switched assignment fails to provide that scalar action in the witness. Its nonzero first-order gap cannot be removed by a modification beginning at second order while the lower-order prescription and identification are retained.

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