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

The representative warning remains

Direct construction of the two new maps › The representative warning remains · lines 864–892

X-C-ST@1#eq:equation-input, X-C-ST@1#eq:input-nondescent

For $a \in D$ ,

s_{P_{m} a} [h, k] i_{P_{m} a} [h] = - L_{2, g} [h, k] a, = L_{1, g} [h] E_{A} L_{1, g} [h] a .

Their classes need not vanish. At the full formal level,

R_{q} (P_{m} a) = P_{m} a - L [q] a + L [q] E_{A} L [q] a (mod κ^{3}) .

Thus neither the formal recursion nor tree-exactness changes the insertion domain into an on-shell quotient.

Non-descent already occurs for the inherited first map. For example, in the massless case with $h = r η$ and a real bump $r$ supported where $g = 1$ , take $a = \partial_{t} r$ . Then

⟨ - L_{1, g} [r η] a, t ⟩ = - \int (\partial_{t} r)^{2} d^{4} x \neq = 0.

This recalls the predecessor's representative test; it is not a new construction of its map.

Displayed formulas in this passage

X-C-ST@1#eq:equation-input · line 867
$s_{P_{m} a} [h, k] i_{P_{m} a} [h] = - L_{2, g} [h, k] a, = L_{1, g} [h] E_{A} L_{1, g} [h] a .$
X-C-ST@1#eq:input-nondescent · line 873
$R_{q} (P_{m} a) = P_{m} a - L [q] a + L [q] E_{A} L [q] a (mod κ^{3}) .$
unlabeled · line 886
$⟨ - L_{1, g} [r η] a, t ⟩ = - \int (\partial_{t} r)^{2} d^{4} x \neq = 0.$