Manuscript XIII-C@1 · XIII-C.tex · section · e6556893

The evaluation boundary and the carrier

C0: copied release and the selected problem › The evaluation boundary and the carrier · lines 355–412

XIII-C@1#eq:inverses, XIII-C@1#eq:HT, XIII-C@1#eq:order, XIII-C@1#eq:witness

From XIII-A:

Under (UCP $_{W}$ ), $Δ_{ξ} O ∣_{C_{W}} = 0$ . The coefficientwise vanishing space $N_{W}$ is not a two-sided ideal. No restricted quantum domain is handed off.

Here

C_{W} = {u \in C_{sol} : D^{*} b = 0 on W}, W \subset Int {g = 1} \cap {ε < t < T} .

This is an evaluation set, not the domain of a new product XIIIA.

The inherited product and relevant inverses are

F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G), G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R},

with $H - H^{T} = i Δ$ XIIA. We use the frozen translation-invariant $H$ , not an unspecified replacement. Its metric–scalar block vanishes. The corresponding time-ordered line is

H_{T} = H + i G_{act}^{A} .

The spacetime is $R^{4}$ , with signature $(+, -, -, -)$ , $□ = \partial_{t}^{2} - Δ_{x}$ , and $P_{m} = □ + m^{2}$ . All external switches and the width $ε > 0$ are fixed. The selected order is

R_{λ V_{g}} O = O + λ (R_{1}^{(0)} + ℏ R_{1}^{(1)}) + O (λ^{2}, λ ℏ^{2}) .

As in XIII-B, $λ = κ$ counts one additional interaction insertion beyond the displayed $κ^{2}$ coefficient.

The copied XII-F value

Δ_{ξ} O (u_{*}) = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0

remains a boundary of the evaluation claim XIIF. It is not a binary anomaly and is not used to choose counterterms.

Displayed formulas in this passage

XIII-C@1#eq:inverses · line 373
$F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G), G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R},$
XIII-C@1#eq:HT · line 384
$H_{T} = H + i G_{act}^{A} .$
XIII-C@1#eq:order · line 393
$R_{λ V_{g}} O = O + λ (R_{1}^{(0)} + ℏ R_{1}^{(1)}) + O (λ^{2}, λ ℏ^{2}) .$
XIII-C@1#eq:witness · line 403
$Δ_{ξ} O (u_{*}) = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0$
unlabeled · line 364
$C_{W} = {u \in C_{sol} : D^{*} b = 0 on W}, W \subset Int {g = 1} \cap {ε < t < T} .$