The comparison assumptions

For clarity, the logical contents of VII-C's four normal-topology extension requirements are reproduced here as comparison requirements, not assumptions of the present theorem VIIC.

Its model includes a quantum metric block $H_{hh}=P{\Delta }_0^{+}$, a scalar block, and a sharp mixed insertion $B_f$ defined with its zero-Cauchy coordinate prescription. Its signature convention is separate from the analytical convention of this file.

A proposed extension in that record must have:

1. a Hausdorff locally convex formal algebra containing the old microcausal polynomial algebra with its fixed Wick product;

2. admitted sharp insertions whose order-${\hslash }^0$ symbols equal the specified $B_f$ on all smooth off-shell configurations;

3. sequentially continuous admission on the specified mixed-kernel normal-topology stages, agreeing with the regular inclusion and sending the prescribed mollified sharp kernels to those insertions;

4. bounded-set hypocontinuous multiplication and a continuous field-independent order-${\hslash }^3$ coefficient extraction, agreeing with the old extraction.

For suitable separated tests, its prescribed word has

where $W_g=\int g{h_{00}}^2{\mathrm{d}}^{4}x$. That is its named extension obstruction, not a claim that every smaller scalar construction is impossible.