Manuscript XV-F@3 · XV-F-v3.tex · section · d4e74afa

The fourth pass: displays from the registered full texts

Source manifest (Gate F0) › The fourth pass: displays from the registered full texts · lines 542–620

XV-F@3#eq:Recrule, XV-F@3#eq:R, XV-F@3#eq:RR, XV-F@3#eq:cross, XV-F@3#eq:hess, XV-F@3#eq:Rconv

module and reconstruction (D0.1–D0.3). XIV-I, def:amodule, lines 873–901 Iamodule: the presentation module is the finite retarded-pole module $A^{m} (V)$ on the adapted chart $w = w_{\pm}$ of (eq:chart) below, reconstructed by

Rec (A) = j, β \sum c_{j β} (w) δ^{(j)} (q) \partial_{r}^{β} R (r), A \in A^{m} (V),

a finite sum, $q = q_{\pm}$ , $r = r_{\pm}$ , with coefficients $c_{j β}$ smooth on the chart and determined by $A$ . The generators of $A^{m} (V)$ and the coefficient class (in particular whether $c_{j β}$ may depend on $q$ , on $r$ , or on both) are described in prose and not reproduced (status P).

kernel (D0.13). XIV-I, eq:R, lines 329–335 IeqR:

R (t, x) = \frac{Θ ( t )}{2 π} δ (t^{2} - ∣ x ∣^{2}),

B := R * R = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .

cross-line operator (D0.11). XIV-J, lines 1112–1123 Jcross: with $π : W \to R^{4} \times R^{4}$ , $π (a, b, z, y) = (y - a, y - b)$ ,

C_{cross} = π^{*} (H^{'}_{γ γ} \otimes H^{'}_{ϕϕ} - H_{γ γ} \otimes H_{ϕϕ}),

the pullback by the submersion $π$ of the difference of two tensor products of blocks written as distributions in the difference variables; $H^{'}$ is this document's placeholder for XIV-J's replaced blocks (F3). Whether these are XII-A's frozen blocks (eq:Hblocks) and an admissible replacement in XII-A's sense is a candidate.

vertex Hessian (D0.12). XIV-J, lines 1128–1145 Jhessian:

V^{''}_{γ ϕ} [h, ψ] = - \int g (y) h_{μν} (y) T^{μν} (ϕ, ψ) (y) d^{4} y,

bilinear in the metric direction $h$ and the scalar direction $ψ$ at the scalar background $ϕ$ , with $g$ the switch. XIV-J calls $T^{μν} (ϕ, ψ)$ the inherited polarized tensor and does not expand it.

convention (D0.15). XIV-J, eq:Rconvention, lines 631–636 Jrconv:

R_{1} (V; O) = \frac{i}{ℏ} (T_{2}^{ren} (V, O) - V ⋆_{H} O) .

This is a map on functionals; it is not the kernel $R$ of (eq:R) (sec:fourR).

{Paraphrased items (status P).} XIV-J, eq:target, lines 720–730 Jtarget: $m^{\circ}$ is “the formal six-branch contraction against the grouped Hadamard difference”; its display is not reproduced. XIV-J, def:mapclass, lines 732–762 Jmapclass: the joint-map class consists of linear maps on reconstructed kernels that (a) contract with the cross-lines, (b) retain both $K_{5}$ placements, (c) agree with the canonical expression on the proved domain, and (d) meet $N$ ; the display and the meaning of (d) are not reproduced. XIV-H's own eq:K9 at line 1055 HK9 is located and not reproduced.

Displayed formulas in this passage

XV-F@3#eq:Recrule · line 549
$Rec (A) = j, β \sum c_{j β} (w) δ^{(j)} (q) \partial_{r}^{β} R (r), A \in A^{m} (V),$
XV-F@3#eq:R · line 562
$R (t, x) = \frac{Θ ( t )}{2 π} δ (t^{2} - ∣ x ∣^{2}),$
XV-F@3#eq:RR · line 566
$B := R * R = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .$
XV-F@3#eq:cross · line 574
$C_{cross} = π^{*} (H^{'}_{γ γ} \otimes H^{'}_{ϕϕ} - H_{γ γ} \otimes H_{ϕϕ}),$
XV-F@3#eq:hess · line 588
$V^{''}_{γ ϕ} [h, ψ] = - \int g (y) h_{μν} (y) T^{μν} (ϕ, ψ) (y) d^{4} y,$
XV-F@3#eq:Rconv · line 600
$R_{1} (V; O) = \frac{i}{ℏ} (T_{2}^{ren} (V, O) - V ⋆_{H} O) .$