Manuscript X-J-ST@1 · X-J-ST.tex · theorem · 8f0d1a36

Theorem: Independent second charge and local matching

Section A: two action-derived compact charged coefficients › Exact local charges and independent values · lines 830–875 · proof included or linked

X-J-ST@1#thm:localmatching, X-J-ST@1#eq:chargematrix, X-J-ST@1#eq:independentflow, X-J-ST@1#eq:commonflow

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The family (eq:witnessfamily) satisfies

C_{i} [ϕ_{b}] = b_{i}, q_{i} (ϕ_{b}; ξ_{j}) = δ_{ij} .

For $∣ t ∣ < t_{*}$ , its scalar-flow comparisons obey

C_{j} [ϕ_{b} \circ F_{t}^{ξ_{i}}] C_{i} [ϕ_{b} \circ F_{t}^{ξ}] = b_{j} + δ_{ij} t, = b_{i} + t .

Consequently

I_{loc} = C_{2} - C_{1}

is invariant on the specified common-flow comparisons.

In particular, the second value can be changed while the first is fixed, and the second charge is not obtained by assigning a compensating number to the first.

Proof. On $K_{i}$ , the cutoffs have $χ_{i} = η_{i} = 1$ , the other cutoffs vanish, and $\partial_{1} ϕ_{b} = 1$ . Equations (eq:moment) and (eq:qi) prove (eq:chargematrix).

A flow trajectory starting in $K_{i}$ , for $∣ t ∣ < t_{*}$ , remains in the region where the relevant vector field equals $\partial_{1}$ . Thus it is exactly $x \mapsto x + t e_{1}$ . The scalar along that trajectory is $x^{1} + t + b_{i}$ . The other supported vector field vanishes there. Integration against $w_{i}$ gives (eq:independentflow)–(eq:commonflow). The difference cancels the common $t$ .

Displayed formulas in this passage

X-J-ST@1#eq:chargematrix · line 833
$C_{i} [ϕ_{b}] = b_{i}, q_{i} (ϕ_{b}; ξ_{j}) = δ_{ij} .$
X-J-ST@1#eq:independentflow · line 839
$C_{j} [ϕ_{b} \circ F_{t}^{ξ_{i}}] C_{i} [ϕ_{b} \circ F_{t}^{ξ}] = b_{j} + δ_{ij} t, = b_{i} + t .$
X-J-ST@1#eq:commonflow · line 839
$C_{j} [ϕ_{b} \circ F_{t}^{ξ_{i}}] C_{i} [ϕ_{b} \circ F_{t}^{ξ}] = b_{j} + δ_{ij} t, = b_{i} + t .$
unlabeled · line 848
$I_{loc} = C_{2} - C_{1}$

Theorem: Independent second charge and local matching

Displayed formulas in this passage

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