Theorem: Sharp Wick-extension incompatibility through 3

There is no Hausdorff locally convex algebra over $\mathbb{C}[\hslash ]/({\hslash }^4)$ satisfying all of:

1. the regular Wick products of the displayed approximants and $W_g$ agree with the old products through ${\hslash }^3$;

2. each unchanged ${\mathcal{B}}_{f,ϵ}$ converges to an admitted sharp ${\mathcal{B}}_f$;

3. multiplication is hypocontinuous with respect to bounded sets;

4. extraction of the field-independent ${\hslash }^3$ coefficient is continuous and has finite complex values.

For a separated positive-residue test with coefficient $C>0$, the violating sequence has the quantitative lower bound

for all sufficiently small $ϵ$.

Proof. Choose the separated tests from Input theorem input:triangle. The approximant families converge by (Q2), hence are bounded. Hypocontinuity implies separate continuity and continuity along pairs of convergent sequences: split the difference of two products and use uniform continuity over the bounded family in the other slot.

Thus

converges. By (Q4), its displayed coefficient converges to a finite number.

By (Q1) it is the old numerical coefficient. Equation (eq:triangle-input) instead gives the stated logarithmic lower bound and divergence. This is a contradiction.