Manuscript XIII-E@1 · XIII-E.tex · section · 74eaba48

The two displayed lifts

E0: the pair, carrier, order, and contact class › The two displayed lifts · lines 445–493

XIII-E@1#eq:observable, XIII-E@1#eq:slots, XIII-E@1#eq:expansion

The only interaction entry is $V_{g}$ of Input statement input:B. The observable is

O = Φ^{(2)}_{D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ), Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D},

with

r_{D} = P E_{0}^{R} J_{D} = - u_{D}, r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]) .

The insertion sign and both scalar-completion legs are unchanged XIIE.

The relevant inverse blocks are

G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R} .

The frozen translation-invariant product data satisfy

H - H^{T} = i (G_{act}^{R} - G_{act}^{A}), H_{T} = H + i G_{act}^{A} .

Metric–scalar cross blocks vanish for these frozen data. All numerical tensor factors and action signs are retained.

The selected expansion is

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

Only the coefficient of $λ ℏ$ is sought. There is no $R_{2}$ , no $T_{3}$ , and no claim of completeness of a higher coupling-order relational coefficient.

The operation remains attached to these off-shell lifts. A representative-independent operation on all of $E_{H}$ is not part of the target XIIIB.

Displayed formulas in this passage

XIII-E@1#eq:observable · line 449
$O = Φ^{(2)}_{D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ), Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D},$
XIII-E@1#eq:slots · line 457
$r_{D} = P E_{0}^{R} J_{D} = - u_{D}, r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]) .$
XIII-E@1#eq:expansion · line 480
$R_{λ V_{g}} O = O + λ (R_{1}^{(0)} + ℏ R_{1}^{(1)}) + O (λ^{2}, λ ℏ^{2}) .$
unlabeled · line 467
$G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R} .$
unlabeled · line 472
$H - H^{T} = i (G_{act}^{R} - G_{act}^{A}), H_{T} = H + i G_{act}^{A} .$