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

Four distinct changes

Finite-time harmonic coordinates and second-order descent › Dependence on the relational prescription · lines 1181–1200

VI@1#fig:VI-prescription-choices

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
1. named diffeomorphism with fixed coordinate data
classical descent
2. change of perturbative gauge-fixing term
3. change of relational coordinate Cauchy data
w↦w+a,□ηa=0w\mapsto w+a,\qquad \Box_\eta a=0O~f[1]−Of[1]=−∫f aμ∂μϕ0\widetilde{\mathcal O}_f^{[1]}-\mathcal O_f^{[1]}=-\int f\,a^\mu\partial_\mu\phi_0generally changes the observable
4. change of solution Cauchy data

No equality arrows among the four operations. All other choices are held fixed in the descent theorem.

Four distinct changes. The descent theorem concerns the named diffeomorphism action with the relational prescription and Cauchy data fixed. Changing coordinate initial data can change the observable already at first order. Gauge fixing, relational prescription, and solution data are therefore not interchangeable choices.

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