Manuscript X-J-ST@1 · X-J-ST.tex · theorem · 8f0d1a36

Theorem: Explicit second coefficient

Section A: two action-derived compact charged coefficients › Finite polarization of the retained vertex · lines 716–751 · proof included or linked

X-J-ST@1#thm:coefficient, X-J-ST@1#eq:moment

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The definition gives

P_{i} [ϕ] = \int a u_{i}^{3} ϕ d^{4} x, C_{i} [ϕ] = \int w_{i} (ϕ - x^{1}) d^{4} x .

In particular, $C_{2}$ is a compact, continuous, ghost-number-zero scalar functional defined by finite Taylor extraction from the X-I-ST interaction.

The two functionals are linearly independent modulo constants. No distributional product, Green operator, or $ℏ$ -dependent coefficient is used.

Proof. The coupling derivative and order extraction in (eq:polarization) leave $V_{a}$ . The coefficient of $z^{3}$ in

\frac{1}{24} \int a (ϕ + z u_{i})^{4} d^{4} x

is $\int a ϕ u_{i}^{3} d^{4} x /6$ . Multiplication by $3!$ proves (eq:moment).

Smooth compact weights give continuity in the compact-open smooth topology. Choose a test variation equal to one near $K_{1}$ and zero near $K_{2}$ ; the derivative vector of $(C_{1}, C_{2})$ is then $(1, 0)$ . Interchanging the supports gives $(0, 1)$ , proving independence.

Displayed formulas in this passage

X-J-ST@1#eq:moment · line 719
$P_{i} [ϕ] = \int a u_{i}^{3} ϕ d^{4} x, C_{i} [ϕ] = \int w_{i} (ϕ - x^{1}) d^{4} x .$
unlabeled · line 738
$\frac{1}{24} \int a (ϕ + z u_{i})^{4} d^{4} x$

Theorem: Explicit second coefficient

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