Manuscript XIV-C@1 · XIV-C.tex · section · a34d13df

The observable naming correction

J0: the pair and extension contract › The observable naming correction · lines 456–498

XIV-C@1#eq:O, XIV-C@1#eq:vertex

The predecessor notation distinguishes

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

from the full displayed coefficient

O = Φ^{(2)}_{D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ) .

The former supplies nine branches; the latter supplies eleven. The target's instruction to retain the scalar-sourced slot and the inherited 48-match inventory therefore fixes the pair $(V_{g}, O)$ , not the nine-branch pair $(V_{g}, Φ^{(2)}_{D})$ .

This is a naming correction made before the calculation, not a change of observable after a detector. Throughout this paper, $Φ^{(2)}_{D}$ keeps its predecessor meaning.

The interaction is the functional

V_{g} = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x, T_{0}^{μν} = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} ((\partial ϕ)^{2} - m^{2} ϕ^{2}) .

It is not an Euler source operator.

The inherited filled slots remain

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 density-weighted source $J_{D}$ and its layer coefficients are those already selected and admitted on the XII/XIII track. No new Einstein–Hilbert insertion is added.

Displayed formulas in this passage

XIV-C@1#eq:O · line 463
$O = Φ^{(2)}_{D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ) .$
XIV-C@1#eq:vertex · line 477
$V_{g} = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x, T_{0}^{μν} = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} ((\partial ϕ)^{2} - m^{2} ϕ^{2}) .$
unlabeled · line 459
$Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D}$
unlabeled · line 490
$r_{D} = P E_{0}^{R} J_{D} = - u_{D}, r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]) .$