Manuscript X-B-ST@1 · X-B-ST.tex · proposition · d03c150c

Proposition: Zero-switch diagnostic

The mixed term, second-order boundary, and diagnostics › Zero switch and a genuine nonzero class · lines 1513–1555 · proof included or linked

X-B-ST@1#prop:zero

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In the separate specialization $g = 0$ ,

R_{1} (0; F_{c}) = 0,

but

O_{κ} = ϕ (f + κ c_{1} [h] + κ^{2} c_{2} [h, k])

retains its coordinate dressing and differs from $ϕ (f)$ in general. The zero-switch plateau algebra is trivial.

Proof. The response vanishes because $L_{1, 0} = 0$ , while (eq:coordinates) does not contain $g$ .

For a nonzero class, take $m = 0$ and real $f \geq 0$ , $f \neq = 0$ . Let $t_{-} = min t (K_{f}) > 0$ , choose $0 < a < b < t_{-}$ , and let $0 \neq = ψ \in C_{c}^{\infty} ((a, b))$ be nonnegative. Choose a compact spatial cutoff equal to one near the spatial projection of $D_{f}$ , and use it to define $h = ψ (t) η$ near the retained past. Finite propagation gives on the observation region

w^{0} = - Ψ (t), w^{i} = 0, Ψ (t) = \int_{0}^{t} σ (s) ψ (s) d s > 0.

Thus $c_{1} = - \partial_{t} (f Ψ)$ , and

⟨ c_{1}, t ⟩ = \int f Ψ d^{4} x > 0.

An equation-ideal element would vanish on the smooth free solution $t$ , so $ϕ (c_{1}) \neq = 0$ .

Finally $int {g = 1}$ is empty at $g = 0$ . This specialization creates no new nonempty gauge algebra.

Displayed formulas in this passage

unlabeled · line 1516
$R_{1} (0; F_{c}) = 0,$
unlabeled · line 1520
$O_{κ} = ϕ (f + κ c_{1} [h] + κ^{2} c_{2} [h, k])$
unlabeled · line 1541
$w^{0} = - Ψ (t), w^{i} = 0, Ψ (t) = \int_{0}^{t} σ (s) ψ (s) d s > 0.$
unlabeled · line 1546
$⟨ c_{1}, t ⟩ = \int f Ψ d^{4} x > 0.$