λEnter Calculus CompanionRegistered manuscripts, kept distinct

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

Named formal gauge action

Action, signs, background, and coefficient domain › The named gauge group · lines 576–594

VI@1#fig:VI-gauge-action

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted

schematic slab

t=Tt=T
supp⁡ξ\operatorname{supp}\xi
supp⁡ζ\operatorname{supp}\zeta
initial surface t=0t=0
Fξ,ζ∗=exp⁡(κLξ+κ2Lζ) mod κ3F_{\xi,\zeta}^*=\exp(\kappa\mathcal L_\xi+\kappa^2\mathcal L_\zeta)\ \mathrm{mod}\,\kappa^3h′=h+Kξh'=h+K\xij′=j+Lξh+Kζ+12Lξ2ηj'=j+\mathcal L_\xi h+K\zeta+\tfrac12\mathcal L_\xi^2\etaϕ0′=ϕ0,ϕ1′=ϕ1+Xξϕ0\phi_0'=\phi_0,\quad \phi_1'=\phi_1+X_\xi\phi_0ϕ2′=ϕ2+Xξϕ1+Xζϕ0+12Xξ2ϕ0\phi_2'=\phi_2+X_\xi\phi_1+X_\zeta\phi_0+\tfrac12 X_\xi^2\phi_0
preserves JT\mathcal J_T
preserves coefficient Cauchy data
substituted residual-wave gauge group

Named formal gauge action. Compact generators supported away from the initial surface induce simultaneous metric and scalar coefficient transformations. Covariance preserves the solution-jet equations, while the support restriction preserves all coefficient Cauchy data. The scalar transformations are essential parts of the action.

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