Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

The actual topology ledger

Three topologies and separate comparison problems › The actual topology ledger · lines 1883–1923

The three named topologies in this comparison are:

the fixed-support smooth topology of $D_{C}$ ;
its image under the continuous map to the free-scalar quotient $Q_{m}$ ;
the different boundary-sensitive topology $τ_{Σ}$ of the IX-A/IX-A-ST actual-functional domain.

Polynomial smoothing growth and characteristic asymptotics are additional comparison estimates, not alternative names for these topologies.

{}1.12

Domain or topology	Object	Verdict and boundary
$D_{C}$	The fixed-background completed scalar tests	Convergent at $O (ε)$ in every seminorm
$Q_{m}$	The same degree-one free classes	Convergent by a continuous quotient map
IX-A-ST's $τ_{Σ}$	Its unchanged sharp functional and regular approximants	Its nonconvergence result is not replaced by this limit
Joint quantum metric kernels	No such limit object is constructed here	No sharp metric-slot admission theorem
Characteristic cross-line composition	A different multi-slot distribution problem	Not an inference from scalar-test convergence