Remark: Status of Gate S-bv

What predecessors supply. XVI-K, eq:S, defines $S:=h_0+i{E}_0^{\mathrm{A}}=-W_0+i{E}_0^{\mathrm{A}}=-{{\Delta }_F}^{\mathrm{FR}}$ through two globally defined distributions, with Fourier $z^0$-family formulae (XVI-K, lem:families / eq:W-fam, eq:E-fam) quoted from XVI-J and XI-E, eq:h ($h_0=-W_0$). Homogeneity of degree $-2$ is proved (XVI-K, lem:homog); $\square S=0$ on $\Omega$ (XVI-K, lem:grading(3)); the Hörmander product of two copies exists on $y\ne w$ (XVI-J / XVI-K, lem:prod). XVI-K states explicitly: “No explicit position-space formula for $S$ is used.” XI-E, eq:W and eq:h, likewise give the Fourier definition of $W_{\mu }$ and the normalization $h_{\mu }=-W_{\mu }$, with no $(Q-i0{)}^{-1}$ formula. XVI-J, however, already prints the position-space formula: eq:DF sets $S=h_0+i{E}_0^{\mathrm{A}}=-d$ with $d=W_0-i{E}_0^{\mathrm{A}}$, and eq:rational states $d(z)=-\frac{1}{4{\pi }^2}(Q_{\mathrm{geom}}(z)-i0{)}^{-1}$ on $z\ne 0$, including nonzero null points; XVI-J, prop:q, uses it (quotations in Section sec:wpA-identify). Its printed justification is “direct Fourier evaluation”.

What was missing. A written real-time derivation of XVI-J, eq:rational, including the boundary-value argument at nonzero null points. The Euclidean Wick calculation of Remark rem:Sbv-motivation motivates the candidate $c=1/(4{\pi }^2)$; it is not a real-time identification of $S$ with $c{T}_1$ on $\Omega$, and it does not settle equality across the nonzero null cone. Lemma lem:bv establishes the model calculus for $T_m$; it does not identify $S$ with $c{T}_1$. No matching probe, collapse rank, or Helmholtz certificate discharges (S-bv).

Certification (this cycle): a written check of XVI-J, eq:rational, not a new extension. Theorem thm:wpA-Sbv establishes, from the predecessor Fourier families, all four required items on $\Omega$: (i) the distributional representation $S=c(Q-i0{)}^{-1}$; (ii) the boundary-value side, the same as for $T_1$ (namely $Q-i0$, not $Q+i0$); (iii) the normalization $c=1/(4{\pi }^2)\ne 0$; (iv) equality across the nonzero null cone, not merely on $\{Q\ne 0\}$. The alternative route (certifying the $185$ relations and their completeness directly) is therefore not needed. Consequently every graded M.2–M.4 (and M.7) conclusion that uses the presentation of Theorem thm:M2 holds unconditionally on $\Omega$; only $c\ne 0$ and the single side were ever used. Statements that also use Gate V remain relative to that chosen convention.