Manuscript XVI-M@1 · XVI-M.tex · section · 7346c909

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

M.2 Numerical-kernel relations and Gate S-bv › Kernel identification: Gate S-bv certified (work package A) · lines 635–665

XVI-M@1#sec:wpA-conventions

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ledger.

Item	Value (predecessor-authoritative)	Source
Difference variable	$z = x - y$ ; $S$ is a translation-invariant distribution of $z$	XVI-K, after eq:S
Caveat	XVI-K uses $z = y - w$ for the bubble extension only; this is not a redefinition of the edge kernel $S$	XVI-K (extension step)
Signature / $Q$	$η = diag (1, - 1, - 1, - 1)$ , $Q (z) = z_{0}^{2} - ∣ z ∣^{2}$	XI-B; Input inp:Sbv
Wave operator	$□ = \partial_{t}^{2} - Δ$ ; $□ E_{0}^{R/A} = id$	XI-B, lem:Green; XVI-J, lem:signs
Spatial Fourier in $W_{0}$	$W_{0} (x, y) = \int \frac{e ^{- i ∣ p ∣ (x^{0} - y^{0}) + i p \cdot (x - y)}}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p$ (massless)	XI-E, eq:W; XVI-K, eq:W-fam
Test-function FT	$φ ~ (z^{0}, p) = \int e^{i p \cdot z} φ d^{3} z$	XVI-K, lem:families
$h_{0}$	$h_{0} = - W_{0}$ , $h_{0} - h_{0}^{T} = i δ_{0}$	XI-E, eq:h
$E_{0}^{A}$ (sine)	$(E_{0}^{A} F) (t) = \int_{t}^{\infty} \frac{s i n (( s - t ) Ω _{0} )}{Ω _{0}} F (s) d s$	XVI-J, eq:sineA; XVI-K, eq:E-fam
$E_{0}^{A}$ (step)	$E_{0}^{A} = - θ (y^{0} - x^{0}) δ_{0} = - θ (- z^{0}) δ_{0}$	XVI-J, eq:step
$S$	$S := h_{0} + i E_{0}^{A} = - W_{0} + i E_{0}^{A} = - Δ_{F}^{FR}$	XVI-K, eq:S; XVI-J, lem:identify
Reconciliation	“ $S = h_{0} + i E_{0}^{A}$ ” and “ $S = - W_{0} + i E_{0}^{A}$ ” are equivalent via $h_{0} = - W_{0}$	XI-E, eq:h; XVI-K, eq:S
Transposition	$⟨ T^{T}, ϕ (x, y)⟩ = ⟨ T, ϕ (y, x)⟩$ ; for $u (z)$ , $u^{T} (z) = u (- z)$	XVI-J: $(E_{0}^{A})^{T} = E_{0}^{R}$ , $W_{0}^{T} = W_{0} + i δ_{0}$
$(Q - i 0)^{- 1}$	$lim_{τ ↓ 0} (Q - i τ)^{- 1}$ in $D^{'} (Ω)$ ; equivalently $Φ^{*} ((s - i 0)^{- 1} \otimes 1)$ near nonzero null points	Lemma lem:bv
One-variable identity	$(s - i 0)^{- 1} = PV (1/ s) + iπ δ (s)$ (sign $+ iπ$ )	classical; checked in `wpA/kernel_ identification_ checks.py`
Constant / side	$c = 1/ (4 π^{2})$ ; boundary side $Q - i 0$ (the side of $T_{1}$ )	Theorem thm:wpA-Sbv

No abstract $R^{4}$ Fourier transform of $S$ is fixed by the predecessors—only the spatial Fourier formulae for $W_{0}$ and the sine kernel. Signs are not inferred from the word “Feynman”; $Δ_{F}^{FR}$ enters only through XVI-J, lem:identify, as an identification of the already-defined combination $θ_{+} W_{0} + θ_{-} W_{0}^{T}$ .

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

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