Kernel identification: Gate S-bv certified (work package A)

-space identification.

What the predecessors already print (audit 3). XI-E, eq:h (box copy enter-companion/raw/tex/XI-E.tex, lines 366–371, verified) only normalizes the Wightman factor and records the bisolution jump:

“To match the proposed block notation put $h_{\mu }=-W_{\mu },h_{\mu }-{h_{\mu }}^{\mathsf{T}}=i{\delta }_{\mu }.$”

It gives no position-space boundary value. XVI-J (version 1, content/manuscripts/XVI-J.tex of cheungngo/enter-companion), eq:W, eq:EA and eq:DF read (macros expanded):

“Define the massless positive-frequency distribution by $W_0(z)={\int }_{{\mathbb{R}}^3}\frac{e^{-i|\mathbf{p}|z^0+i\mathbf{p}\cdot \mathbf{z}}}{(2\pi {)}^{3}2|\mathbf{p}|}{\mathrm{d}}^3\mathbf{p}.$ The integral is distributional. Set $h_0=-W_0$.” \ “${\widehat{E_0}}^{\mathrm{A}}(t,\mathbf{p})=-\theta (-t)\frac{\sin (t|\mathbf{p}|)}{|\mathbf{p}|}$” \ “Put $d=W_0-i{E}_0^{\mathrm{A}},S=h_0+i{E}_0^{\mathrm{A}}=-d,D_F=-\mathcal{P}S=\mathcal{P}d.$”

XVI-J, eq:rational:

“Let $Q_{\mathrm{geom}}(z)=(z^0{)}^2-|\mathbf{z}{|}^2.$ On $z\ne 0$, direct Fourier evaluation gives $d(z)=-\frac{1}{4{\pi }^2}(Q_{\mathrm{geom}}(z)-i0{)}^{-1}.$ At a nonzero null point, $Q_{\mathrm{geom}}$ is a submersion. The boundary value in [eq:rational] has one, rather than both, conormal orientations.”

XVI-J, prop:q:

“The distribution $q^{\circ }(z)=S(z{)}^2=d(z{)}^2,z\ne 0,$ exists on ${\mathbb{R}}^4\setminus \{0\}$, including nonzero null points. It is homogeneous of degree $-4$, has scaling degree four at the origin, and is nonzero.”

Thus $S=-d=\frac{1}{4{\pi }^2}(Q-i0{)}^{-1}$ on $z\ne 0$, including nonzero null points, is the XVI-J formula ($Q=Q_{\mathrm{geom}}$), and XVI-J, prop:q, uses it. Theorem thm:wpA-Sbv is a written check of XVI-J, eq:rational, whose printed justification is “direct Fourier evaluation”; it is not a new extension. XVI-M's contribution is (i) the written real-time derivation $W_0=-\frac{1}{4{\pi }^2}(\mathrm{PV}\frac{1}{Q}+i\pi \mathrm{sgn}(z^0)\delta (Q))$, ${E_0}^{\mathrm{A}}=\frac{1}{2\pi }\theta (-z^0)\delta (Q)$ and the cancellation giving $\frac{1}{4{\pi }^2}(Q-i0{)}^{-1}$; and (ii) the local justification at nonzero null points of the boundary value of the variable regulator $Q-2i\tau z^0-{\tau }^2$ (Lemma lem:wpA-varreg). Predecessor provenance. Each predecessor version used below is pinned by SHA-256 (local files) or by an archived corpus excerpt under XVI-M-artifacts/ provenance/ (regression re-checks those hashes). Citations are by label only.

• Local XVI-J.tex (dcba2602ab4a): labels eq:W0, eq:sineA, eq:step, lem:signs, lem:identify. Role: supplementary (earlier box version).

• Repository fetch of XVI-J (d5558cd419c8, provenance/XVI-J_github_ fetch.tex): HTTP 404 stub. Role: failed retrieval; does not remove the corpus file.

• Corpus excerpt of XVI-J ({eda87cb7400dł