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

Theorem: Local observation-diagonal limitation

Residue geometry and the observation-diagonal limitation · lines 910–960 · proof included or linked

IX-B-ST@1#thm:diagonal, IX-B-ST@1#eq:log

There exist nonnegative $f, g, f^{'} \in D (V; R)$ with pairwise disjoint supports such that

C (f, g, f^{'}) := ⟨ r, f \otimes g \otimes f^{'} ⟩ > 0.

For those tests,

⟨ T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ = C (f, g, f^{'}) lo g (ε^{- 1}) + O (1) .

No family of counterterms supported on $Δ_{12} \cup Δ_{23} \cup Δ_{13}$ removes this divergence on all joint tests. The same conclusion holds for observation faces avoided by the chosen compact supports.

Proof. Choose

0 < λ < min {T_{0} /4, r_{0} /5}

and one unit vector $n_{0}$ . The relation contains the triple

y_{0} = (λ, 0), x_{0} = (2 λ, 3 λ n_{0}), x^{'}_{0} = (3 λ, 4 λ n_{0}) .

All three points lie in $V$ and are distinct. Take nonnegative bumps with mutually disjoint sufficiently small supports, positive near these points. Their product is positive on an open parameter neighbourhood of this triple. Lemma lem:measure gives the positive residue pairing.

Assumption ass:VIID gives the logarithmic term plus a finite distributional pairing and a vanishing remainder. The contribution $- lo g μ C$ is constant at fixed $μ$ , proving (eq:log).

The product-test support avoids every observation equality diagonal. Every distribution supported on their union pairs to zero with it, even when the distribution depends on $ε$ . Such counterterms cannot cancel the nonzero logarithm.

Displayed formulas in this passage

IX-B-ST@1#eq:log · line 919
$⟨ T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ = C (f, g, f^{'}) lo g (ε^{- 1}) + O (1) .$
unlabeled · line 914
$C (f, g, f^{'}) := ⟨ r, f \otimes g \otimes f^{'} ⟩ > 0.$
unlabeled · line 933
$0 < λ < min {T_{0} /4, r_{0} /5}$
unlabeled · line 938
$y_{0} = (λ, 0), x_{0} = (2 λ, 3 λ n_{0}), x^{'}_{0} = (3 λ, 4 λ n_{0}) .$

Theorem: Local observation-diagonal limitation

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