Latest supplied XVI-E/F results

The supplied XVI-E retains $T^{\mathbb{C}}=Q_{\lambda }$ through the real root-theorem values and proves fixed-data rapid decay of that scalar root pairing XVIE.

The supplied XVI-F-v3.tex goes further than the earlier XVI-F versions:

• it constructs $I_{\mathrm{cand}}$ independently;

• it proves $I_{\mathrm{cand}}=Q_{\lambda }$ in prop:compat;

• it identifies the advanced root on the slab using the reported smooth-source sine-integral definition;

• it proves direct fixed-data rapid decay of $I_{\mathrm{cand}}$;

• it does not identify the auxiliary coordinate field with the inherited retarded coordinate prescription.

The earlier unresolved residual is therefore not repeated as the current result of XVI-F-v3. That version proves vanishing of the residual pairing; it does not require the discrepancy tensor to vanish at every negative time XVIFvThree.

These are later results on named scalar evaluations. They are not premises of the response proof in Section sec:response. They do not by themselves prove the retarded operator identity in Proposition prop:coordinate-compatibility.

Their rapid-decay assertions hold at fixed regulator and fixed real smooth $\varphi$ independent of $\lambda$. They are not pointwise response decay, a frequency-dependent scalar estimate, or a class-admission theorem.