Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

The inherited first response and the unfilled slot

Frozen analytical data and predecessor handoff › The inherited first response and the unfilled slot · lines 603–640

X-C-ST@1#eq:first-map, X-C-ST@1#eq:rho2

The first map is recalled, not rebuilt:

R_{1} (V_{1, g} [h]; F_{c}) = ϕ (- L_{1, g} [h] E_{A} c) .

X-B-ST supplies its support, continuity, tree-exactness, and first-order completed identity, as well as

A_{1} [h] = ϕ (c_{1} [h] - L_{1, g} [h] E_{A} f) .

The classical target for the additional second-order slot is the supplied expression

ρ_{2, g} [h, k; u] = E_{R} L_{1, g} [h] E_{R} L_{1, g} [h] u - E_{R} L_{2, g} [h, k] u .

For free $u$ , it is the second scalar Cauchy coefficient of X-A-ST. For arbitrary smooth $u$ , the same formula is a well-defined smooth linear extension, because every source operator has compact coefficients.

The predecessor's incomplete second coefficient was

ϕ (c_{2} [h, k]) + R_{1} (V_{1, g} [h]; F_{c_{1} [h]}) .

It did not include (eq:rho2). The new pair below fills that one slot.

Displayed formulas in this passage

X-C-ST@1#eq:first-map · line 606
$R_{1} (V_{1, g} [h]; F_{c}) = ϕ (- L_{1, g} [h] E_{A} c) .$
X-C-ST@1#eq:rho2 · line 620
$ρ_{2, g} [h, k; u] = E_{R} L_{1, g} [h] E_{R} L_{1, g} [h] u - E_{R} L_{2, g} [h, k] u .$
unlabeled · line 613
$A_{1} [h] = ϕ (c_{1} [h] - L_{1, g} [h] E_{A} f) .$
unlabeled · line 633
$ϕ (c_{2} [h, k]) + R_{1} (V_{1, g} [h]; F_{c_{1} [h]}) .$