Manuscript IX-E-2@1 · IX-E-2.tex · section · e70d9077

VII-C and the finite part are not repaired

Explicit inherited inputs and their limits › VII-C and the finite part are not repaired · lines 426–459

IX-E-2@1#eq:old-residue, IX-E-2@1#eq:numerical-finite-part

Under VII-C's joint hypotheses (E1)–(E4), namely the old Wick product on the polynomial domain, off-shell admission of the sharp insertion, sequential normal-topology admission, and hypocontinuous multiplication with a continuous $ℏ^{3}$ coefficient map, the mandatory word has no sequentially continuous extension. Its supplied separated-support residue is

C (f, g, f^{'}) = \frac{1}{64 ( 2 π ) ^{3}} \int g (y) d^{4} y \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n),

where $A_{f}$ denotes the inherited reflected light-ray transform. This residue is not subtracted here.

For the supplied VII-D regularization family,

u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r, r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}},

is a strong distributional limit, and admissible mollifiers differ by $c_{ab} r$ . This is numerical finite-part data, not an algebra construction. Neither $u_{μ}^{ρ} + cr$ nor the triangle scalar is renamed as $⋆_{c}$ or $T_{2}$ .

Displayed formulas in this passage

IX-E-2@1#eq:old-residue · line 434
$C (f, g, f^{'}) = \frac{1}{64 ( 2 π ) ^{3}} \int g (y) d^{4} y \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n),$
IX-E-2@1#eq:numerical-finite-part · line 446
$u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r, r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}},$