Manuscript III-B@1 · III-B-v1.tex · theorem · d2c98f62

Theorem: Nonzero compact coupled quotient observable

Compact coupled Ward observables › A strictly positive compact pair · lines 1458–1494 · proof included or linked

III-B@1#thm:compact-pair, III-B@1#eq:positive

The pair (eq:positive-pair) satisfies (eq:Ward). For the supported solution $(h, φ) = (0, Φ)$ ,

O_{J, q} (0, Φ) = 2 (\int_{s_{1}}^{s_{2}} a_{0} (s) csc^{2} (m s) d s) (\int b (x) d^{3} x) > 0.

Moreover $q \in / P C_{c}^{\infty} (L)$ .

Proof. The coefficient support lies in the plateau (eq:plateau). There

\partial_{i} Φ = 0, \partial^{0} Φ = A_{*} m sin (m s) .

The temporal Ward component is

- 2 a_{0}^{'} b + q A_{*} m sin (m s) = 0;

the spatial components vanish.

Since $h = 0$ , $φ = Φ$ solves the triangular equations. Its pairing is

\int q Φ d^{4} x = \frac{2}{m} (\int a_{0}^{'} (s) cot (m s) d s) \int b .

Integrating by parts gives (eq:positive). If $q = P ψ$ for compact smooth $ψ$ , then $\int q Φ = \int ψ P Φ = 0$ , a contradiction.

Displayed formulas in this passage

III-B@1#eq:positive · line 1462
$O_{J, q} (0, Φ) = 2 (\int_{s_{1}}^{s_{2}} a_{0} (s) csc^{2} (m s) d s) (\int b (x) d^{3} x) > 0.$
unlabeled · line 1475
$\partial_{i} Φ = 0, \partial^{0} Φ = A_{*} m sin (m s) .$
unlabeled · line 1480
$- 2 a_{0}^{'} b + q A_{*} m sin (m s) = 0;$
unlabeled · line 1487
$\int q Φ d^{4} x = \frac{2}{m} (\int a_{0}^{'} (s) cot (m s) d s) \int b .$