Manuscript VII-D@1 · VII-D.tex · theorem · ef67c89c

Theorem: Complete regulator conversion

Complete conversion and the real counterterm class · lines 1945–1985 · proof included or linked

VII-D@1#thm:conversion, VII-D@1#eq:conversion

The difference has the strong limit

L_{ab} = ε ↓ 0 lim (T_{ε}^{a} - T_{ε}^{b}) = c_{ab} r, c_{ab} = \int_{0}^{\infty} \frac{ϑ _{a} ( u ) ^{4} - ϑ _{b} ( u ) ^{4}}{u} d u \in R .

For every common scale $μ > 0$ ,

u_{μ}^{a} - u_{μ}^{b} = L_{ab} .

This is the complete conversion distribution: no further normal derivative, nonconstant weight on $Rel$ , or additional term supported on $Rel \cap Δ_{13}$ occurs.

Proof. The distribution $U_{m}$ in (eq:finitepart) is independent of the profile. Hence

L_{ab} = (κ_{ρ_{a}} - κ_{ρ_{b}}) r .

Combining the two integrals in (eq:kappa) gives the convergent integral in (eq:conversion). It is real because both profiles are real.

Applying (eq:finalrate) to both regulators proves strong convergence of their complete difference on all joint tests. This leaves no distribution on a smaller observation stratum undetected.

Displayed formulas in this passage

VII-D@1#eq:conversion · line 1948
$L_{ab} = ε ↓ 0 lim (T_{ε}^{a} - T_{ε}^{b}) = c_{ab} r, c_{ab} = \int_{0}^{\infty} \frac{ϑ _{a} ( u ) ^{4} - ϑ _{b} ( u ) ^{4}}{u} d u \in R .$
unlabeled · line 1963
$u_{μ}^{a} - u_{μ}^{b} = L_{ab} .$
unlabeled · line 1974
$L_{ab} = (κ_{ρ_{a}} - κ_{ρ_{b}}) r .$

Nearby in this section

section: Complete conversion and the real counterterm class