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

Classical descent

Finite-time harmonic coordinates and second-order descent › The observable · lines 1134–1152

VI@1#fig:VI-classical-descent

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
physical coordinates xx
relational labels XX
Xγ′=Xγ∘F,Yγ′=F−1∘Yγ,ϕ′=ϕ∘FX_{\gamma'}=X_\gamma\circ F,\qquad Y_{\gamma'}=F^{-1}\circ Y_\gamma,\qquad \phi'=\phi\circ Fϕ′∘Yγ′=ϕ∘F∘F−1∘Yγ=ϕ∘Yγ\phi'\circ Y_{\gamma'}=\phi\circ F\circ F^{-1}\circ Y_\gamma=\phi\circ Y_\gamma
Of=∫f(X) ϕ(Yγ(X)) d4X\mathcal O_f=\int f(X)\,\phi(Y_\gamma(X))\,\mathrm d^4X
ψ1=ϕ1−Xwϕ0\psi_1=\phi_1-X_w\phi_0ψ2=ϕ2−Xwϕ1+(wν∂νwμ−vμ)∂μϕ0+12wμwν∂μ∂νϕ0\psi_2=\phi_2-X_w\phi_1+(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0+\tfrac12 w^\mu w^\nu\partial_\mu\partial_\nu\phi_0
formal through order two
fixed relational-label smearing
classical covariance, not quantum renormalizability

Classical descent by coordinate covariance. The inverse coordinate map compensates the pullback of the scalar, leaving the relational scalar unchanged through second order. The proof applies formally off shell and restricts to the named solution-jet quotient. It does not establish the quantum products or renormalizability of the composite.

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