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

Three coefficient constructions

Bare dressing, solution coefficients, and external response › Why the first-order matching claim needs qualification · lines 1373–1391

VI@1#fig:VI-three-coefficients

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
bare parenthesesOf(0),Of(1),Of(2)\mathcal O_f^{(0)},\mathcal O_f^{(1)},\mathcal O_f^{(2)}on independent (h,φ)(h,\varphi)
solution-jet bracketsOf[0],Of[1],Of[2]\mathcal O_f^{[0]},\mathcal O_f^{[1]},\mathcal O_f^{[2]}after h↦h+κjh\mapsto h+\kappa j and φ↦ϕ0+κϕ1+κ2ϕ2\varphi\mapsto\phi_0+\kappa\phi_1+\kappa^2\phi_2
switched candidate∫fϕ1,g−∫fXw[h]ϕ0\int f\phi_{1,g}-\int f X_{w[h]}\phi_0
v[h,j]=w[j]+v0[h]v[h,j]=w[j]+v_0[h]
Forced and dressing terms stay distinct. No double counting of sourced jj. The external response alone is not the complete relational coefficient.

Three coefficient constructions. Bare dressing coefficients are functionals of independent fields; solution-jet coefficients include the dynamical scalar and metric corrections. The proposed switched first coefficient combines an external forced response with a geometric dressing. Identifying only one contribution with the complete coefficient omits terms or double counts dynamical corrections.

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