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

Assumption: VII-D parent finite-part input

Chart formulae and the exact dependency ledger › Parent finite part · lines 622–676

IX-B-ST@1#ass:VIID, IX-B-ST@1#eq:parent, IX-B-ST@1#eq:parentestimate, IX-B-ST@1#eq:parentconversion, IX-B-ST@1#eq:parentscale

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The prescribed family (eq:triangle) is smooth for positive cutoff and agrees with the fully contracted old coefficient on product tests. There is a regulator-independent distribution $U_{m} \in D^{'} (X)$ such that

u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r,

where

κ_{ρ} = \int_{0}^{1} \frac{ϑ _{ρ} ( v ) ^{4} - 1}{v} d v + \int_{1}^{\infty} \frac{ϑ _{ρ} ( v ) ^{4}}{v} d v .

For every compact $A ⋐ X$ ,

⟨ T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}, Φ ⟩ \leq C_{A, ρ, m} ε (1 + ∣ lo g ε ∣) p_{A, 64} (Φ), Φ \in D_{A} (X) .

The range is $0 < ε \leq 1/2$ , further restricted when required by the supported smoothing margin. The estimate includes compact sets meeting $R \cap Δ_{13}$ , where $Δ_{13} = {x = x^{'}}$ .

For admissible regulators $ρ_{a}, ρ_{b}$ ,

u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}} c_{ab} + c_{b c} = c_{ab} r, = c_{a c}, c_{ab} u_{μ_{2}}^{ρ} - u_{μ_{1}}^{ρ} = \int_{0}^{\infty} \frac{ϑ _{ρ_{a}} ( v ) ^{4} - ϑ _{ρ_{b}} ( v ) ^{4}}{v} d v \in R, = lo g (μ_{2} / μ_{1}) r .

The regulator conversion is complete within this fixed parent class: no extra normal derivative, nonconstant weight, or independent intersection-supported term occurs. These are the specific VII-D inputs CheungVIID ({:finitepart}, {:conversion}, {:placement}, {:positive}).

Displayed formulas in this passage

IX-B-ST@1#eq:parent · line 629
$u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r,$
IX-B-ST@1#eq:parentestimate · line 640
$⟨ T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}, Φ ⟩ \leq C_{A, ρ, m} ε (1 + ∣ lo g ε ∣) p_{A, 64} (Φ), Φ \in D_{A} (X) .$
IX-B-ST@1#eq:parentconversion · line 657
$u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}} c_{ab} + c_{b c} = c_{ab} r, = c_{a c}, c_{ab} u_{μ_{2}}^{ρ} - u_{μ_{1}}^{ρ} = \int_{0}^{\infty} \frac{ϑ _{ρ_{a}} ( v ) ^{4} - ϑ _{ρ_{b}} ( v ) ^{4}}{v} d v \in R, = lo g (μ_{2} / μ_{1}) r .$
IX-B-ST@1#eq:parentscale · line 657
$u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}} c_{ab} + c_{b c} = c_{ab} r, = c_{a c}, c_{ab} u_{μ_{2}}^{ρ} - u_{μ_{1}}^{ρ} = \int_{0}^{\infty} \frac{ϑ _{ρ_{a}} ( v ) ^{4} - ϑ _{ρ_{b}} ( v ) ^{4}}{v} d v \in R, = lo g (μ_{2} / μ_{1}) r .$
unlabeled · line 634
$κ_{ρ} = \int_{0}^{1} \frac{ϑ _{ρ} ( v ) ^{4} - 1}{v} d v + \int_{1}^{\infty} \frac{ϑ _{ρ} ( v ) ^{4}}{v} d v .$

Assumption: VII-D parent finite-part input

Displayed formulas in this passage

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