Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

Not the unchanged multiplied-equation prescription

Identification: continuous coordinates and normal flux › Not the unchanged multiplied-equation prescription · lines 1344–1385

X-D-ST@1#eq:homogeneous-difference, X-D-ST@1#eq:multiplied-kernel

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Let $w_{0}$ solve

□ w_{0}^{ρ} = C^{ρ} [h], w_{0}^{ρ} (0) = \partial_{t} w_{0}^{ρ} (0) = 0.

Then

w_{0}^{ρ} - w_{0}^{ρ} = - S (t) a_{Σ}^{0 ρ} .

Both solve the same positive-time bulk equation, but have different velocities.

The difference is also visible after transposition. For $n = d t$ , define

B_{Σ} (b)^{α β} = - n^{(α} b^{β)} + \frac{1}{2} η^{α β} n_{μ} b^{μ} .

The multiplied-equation kernel is

C^{*} (θ b) = θ C^{*} b + δ_{Σ} B_{Σ} (b ∣_{Σ}),

whereas (eq:sharp-kernel) lacks the last term.

These statements identify which sharp prescription is obtained. They do not establish sharp joint microcausal admission in quantum metric slots. The corresponding source distinction is the one recorded in X-D XD (:identification).

Displayed formulas in this passage

X-D-ST@1#eq:homogeneous-difference · line 1352
$w_{0}^{ρ} - w_{0}^{ρ} = - S (t) a_{Σ}^{0 ρ} .$
X-D-ST@1#eq:multiplied-kernel · line 1368
$C^{*} (θ b) = θ C^{*} b + δ_{Σ} B_{Σ} (b ∣_{Σ}),$
unlabeled · line 1347
$□ w_{0}^{ρ} = C^{ρ} [h], w_{0}^{ρ} (0) = \partial_{t} w_{0}^{ρ} (0) = 0.$
unlabeled · line 1362
$B_{Σ} (b)^{α β} = - n^{(α} b^{β)} + \frac{1}{2} η^{α β} n_{μ} b^{μ} .$