Manuscript XIV-I@1 · XIV-I.tex · section · 29df501d

Both internal placements of K 5

I0: copied inputs and the exact predecessor stop › Both internal placements of K 5 · lines 509–611

XIV-I@1#input:E, XIV-I@1#eq:regroup, XIV-I@1#eq:B, XIV-I@1#eq:root, XIV-I@1#input:DC, XIV-I@1#input:BC

For smooth directions,

M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} \partial_{α} (σ_{ε} B^{α μ} [G; h]) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x,

where

C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),

B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}], r_{h} = P E_{0}^{R} (J_{D}^{'} [G] h) .

The complete scalar root is

\int f p [r_{h}; ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x, F = E_{m}^{A} f .

These are exact precomposition identities, not characteristic products. The tested attachment bound is not a uniform untested composition theorem XIVE.

On the retained patch,

K_{ret} = K_{4} + K_{5} + K_{6} + K_{7} + K_{8} + K_{9} .

Both gravitational-response legs are retained. The numerator $Q λ$ belongs only to the leading differential fiber calculation for $B_{6}$ XIVD.

The common space is

W = A \times B \times Z \times Y, (a, b, z, y),

with the residual variable $z$ retained. The characteristic test locus is

N = N_{att} = {a = z, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β}, ℓ_{n} = (1, n) .

It has codimension eight in $W$ .

The three separated cone conormals have the copied positive relation. Defined interaction collision contacts vanish on the retained collision-free attachment neighborhood XIVC,XIVD.

XIII-C's four obligations remain:

a characteristic prescription or complete-pair cancellation;
uniform transverse estimates at three- and four-vertex collision strata and their intersections;
compatible partial contacts with singular spectators;
output wavefront control after proper pushforward.

The parallel temporal kinetic primitive has collision scaling degree eight in codimension four XIIIC.

XIII-B supplies the admitted tree datum and the convention

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

The strictly ordered full-support values are zero or the scaled commutator. Properness does not define an absent distributional composition XIIIB.

Displayed formulas in this passage

XIV-I@1#eq:regroup · line 514
$M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} \partial_{α} (σ_{ε} B^{α μ} [G; h]) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x,$
XIV-I@1#eq:B · line 532
$B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}], r_{h} = P E_{0}^{R} (J_{D}^{'} [G] h) .$
XIV-I@1#eq:root · line 545
$\int f p [r_{h}; ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x, F = E_{m}^{A} f .$
unlabeled · line 527
$C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),$
unlabeled · line 561
$K_{ret} = K_{4} + K_{5} + K_{6} + K_{7} + K_{8} + K_{9} .$
unlabeled · line 569
$W = A \times B \times Z \times Y, (a, b, z, y),$
unlabeled · line 574
$N = N_{att} = {a = z, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β}, ℓ_{n} = (1, n) .$
unlabeled · line 602
$R_{1} (V_{g}; O) = \frac{i}{ℏ} (T_{2}^{ren} (V_{g}, O) - V_{g} ⋆_{H} O) .$

Both internal placements of K 5

Displayed formulas in this passage

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