Manuscript IX-B-ST@1 · IX-B-ST.tex · corollary · f6af1523

Corollary: Descendant conversion and scale dependence

Observation derivatives and smooth spectators · lines 1068–1090 · proof included or linked

IX-B-ST@1#cor:conversion, IX-B-ST@1#eq:descconversion

For the same operation,

a D (u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}}) a D (u_{μ_{2}}^{ρ} - u_{μ_{1}}^{ρ}) a D (T_{ε}^{ρ_{a}} - T_{ε}^{ρ_{b}}) = c_{ab} a D r, = lo g (μ_{2} / μ_{1}) a D r, ⟶ c_{ab} a D r strongly .

The last convergence has the summed order- $64 + d$ remainder bound of the two parent estimates.

Proof. The first two identities follow by applying the linear operation $a D$ to (eq:parentconversion)–(eq:parentscale). Subtracting the two instances of (eq:descendantrate) proves the last assertion, since the leading logarithmic terms agree.

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IX-B-ST@1#eq:descconversion · line 1071
$a D (u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}}) a D (u_{μ_{2}}^{ρ} - u_{μ_{1}}^{ρ}) a D (T_{ε}^{ρ_{a}} - T_{ε}^{ρ_{b}}) = c_{ab} a D r, = lo g (μ_{2} / μ_{1}) a D r, ⟶ c_{ab} a D r strongly .$

Corollary: Descendant conversion and scale dependence

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