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

Linearized observable quotient

The linearized interface and a sourced nonempty domain › What linearization at the chosen background gives · lines 672–695

VI@1#fig:VI-linear-quotient

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted

\mathfrak g

(a,u)

\mathcal K_0

(-2G^{(1)}(a),Pu)

\mathcal P_0

$\mathcal K_0\xi=(K\xi,0)$

\mathcal K_0^*(J,c)=0\Longleftrightarrow \partial_\mu J^{\mu\nu}=0

quotient by

\operatorname{im}\mathcal P_0

, and

\mathcal K_0^*\mathcal P_0=0

Numerator tests annihilate pure gauges. Denominator tests annihilate free solutions.

algebraic gauge-invariant linear functionals only

complete continuous-observable classification

specified radical

nonlinear relational observable

Linearized observable interface at $(η, 0)$ . The scalar entry of the linearized gauge map vanishes for the chosen zero-scalar background. Compact conserved tensor tests annihilate pure gauges, and field-equation tests annihilate free solutions. The quotient supplies linear functionals, not a classification of the nonlinear relational observable or its quantum realization.

Linearized observable quotient

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