Manuscript XV-E@1 · XV-E.tex · section · a3dc45f5

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Frozen ledger · lines 441–525

XV-E@1#eq:retained, XV-E@1#eq:K5a, XV-E@1#eq:K5z, XV-E@1#eq:K47, XV-E@1#eq:K6, XV-E@1#eq:pole, XV-E@1#eq:reference, XV-E@1#eq:referenceWF

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The retained kernel and supported sector are

K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9}, K_{5} = K_{5}^{[a]} + K_{5}^{[z]}, K_{supp} = K_{6} + K_{8} + K_{9} .

$K_{47}$ groups $B_{4} + B_{7}$ ; both $K_{5}$ placements belong to $B_{5}$ . The supplied internal expressions are Hkernel,Iamplitude

K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],

and the internal pole chart is

M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, ρ > 0, ∣ ω - s ρ ∣ < c ρ, 0 < c < 1,

with $(λ - i 0)^{- 1} = PV \frac{1}{λ} + iπ δ (λ)$ Ipole. The supported-sector reference and its wavefront bound are Jreference

t_{D}^{\circ} = ∣ α ∣ \leq 3 β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} (v_{α} c_{D, β}) on U ∖ N_{U},

WF (t_{D}^{\circ}) \subseteq Γ^{\circ}_{supp} = N^{*} S_{resp}_{U ∖ N_{U}} ∖ 0 \cup i = 1 ⋃^{2} {(w, λ d q_{i} + ω) : w \in S_{resp} ∖ N_{U}, q_{i} (w) = 0, λ \neq = 0, ω \in N_{w}^{*} S_{resp}} ∖ 0.

Entry	Internal expression	Cross-line factors	Status at $N$
$K_{47}$	(eq:K47); $L$ candidate only	$C_{cross}$ undefined	Composed on $U_{can}$ only Ijoint; not defined on tests meeting $N$ .
$K_{5}^{[a]}$	(eq:K5a); pullback existence proved Hkernel,XVB	Undefined	I3 stop Ijoint.
$K_{5}^{[z]}$	(eq:K5z); pullback existence proved	Undefined	I3 stop.
$K_{6}$	(eq:K6); $D$ via undefined $a ˙_{2}$	Traces $c_{D, β}$ named	Conditional interface only Jconditional.
$K_{8}$ , $K_{9}$	Reported, not reproduced	Traces named	Conditional interface only.

This is a freeze check, not evidence that the full operation exists. Propagator, pole, regulator, carrier, normalization, and ambiguity class are unchanged.

Displayed formulas in this passage

XV-E@1#eq:retained · line 444
$K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9}, K_{5} = K_{5}^{[a]} + K_{5}^{[z]}, K_{supp} = K_{6} + K_{8} + K_{9} .$
XV-E@1#eq:K5a · line 452
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-E@1#eq:K5z · line 452
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-E@1#eq:K47 · line 452
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-E@1#eq:K6 · line 452
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-E@1#eq:pole · line 472
$M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, ρ > 0, ∣ ω - s ρ ∣ < c ρ, 0 < c < 1,$
XV-E@1#eq:reference · line 481
$t_{D}^{\circ} = ∣ α ∣ \leq 3 β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} (v_{α} c_{D, β}) on U ∖ N_{U},$
XV-E@1#eq:referenceWF · line 488
$WF (t_{D}^{\circ}) \subseteq Γ^{\circ}_{supp} = N^{*} S_{resp}_{U ∖ N_{U}} ∖ 0 \cup i = 1 ⋃^{2} {(w, λ d q_{i} + ω) : w \in S_{resp} ∖ N_{U}, q_{i} (w) = 0, λ \neq = 0, ω \in N_{w}^{*} S_{resp}} ∖ 0.$