λEnter Calculus CompanionRegistered manuscripts, kept distinct

Manuscript III@2 · III.tex · figure · 82dad3e5

Four descent problems

Introduction: the equation space precedes descent › The four results · lines 296–320

III@2#fig:descent-ledger

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metricscalarquotient / invariantobstruction / errorsupport / scope
Equation space first; descent second.
declared variables and equations
identification or proposed transformation
result
G(1)(h)=0,Pϕ=0G^{(1)}(h)=0,\quad P\phi=0
q0(h,ϕ)=([h],ϕ)q_0(h,\phi)=([h],\phi)
Exhibited Poisson symbol does not descend.
Pϕ=0,δϕ=κXξϕP\phi=0,\quad \delta\phi=\kappa X_\xi\phi
proposed action on the free scalar shell\text{proposed action on the free scalar shell}
Preserves every compact-data solution exactly when ξ is Killing.\text{Preserves every compact-data solution exactly when }\xi\text{ is Killing.}
Pϕ0=0,Pϕ1=−Lhϕ0P\phi_0=0,\quad P\phi_1=-L_h\phi_0
(h,ϕ0,ϕ1)↦(h+Kξ,ϕ0,ϕ1+Xξϕ0)(h,\phi_0,\phi_1)\mapsto(h+K\xi,\phi_0,\phi_1+X_\xi\phi_0)
Complete functional descends for the specified retarded gauge group.
Φ fixed,G(1)(h)=0,Pφ=−LhΦ\Phi\text{ fixed},\quad G^{(1)}(h)=0,\quad P\varphi=-L_h\Phi
C1 triangular system\text{C1 triangular system}
Hausdorff quotient Cauchy problem; nonzero compact Ward observables.

No arrow identifies these four quotients. They use different variables, equations, and actions.

classical descent
representation · not established here
instrument · not established here
record · not established here

The four rows are different equation-space questions. The second row does not define a quotient by general compact gauges. No arrow identifying the rows is presumed.

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