Manuscript XVI-H@1 · XVI-H.tex · section · af895076

The resonant probes are not an on-shell calculation

On-shell, gauge, and quantum boundaries › The resonant probes are not an on-shell calculation · lines 1580–1603

With $ω = 2 λ$ ,

P_{m} (b cos ω t) P_{m} (b sin ω t) = (P_{m} b - ω^{2} b) cos ω t - 2 ω (\partial_{t} b) sin ω t, = (P_{m} b - ω^{2} b) sin ω t + 2 ω (\partial_{t} b) cos ω t .

Current XVI-G supplies these exact identities and explains off-shell behavior for fixed nonzero $b$ at sufficiently large frequency. The scalar probes in (eq:res-cos)–(eq:res-sin) are not the free scalar solutions quantified in (eq:Isol).

No replacement by free solutions is shown to preserve $Υ [b]$ . Thus witness matching transfers the stated off-shell scalar evaluation, not an on-shell asymptotic.

Displayed formulas in this passage

unlabeled · line 1583
$P_{m} (b cos ω t) P_{m} (b sin ω t) = (P_{m} b - ω^{2} b) cos ω t - 2 ω (\partial_{t} b) sin ω t, = (P_{m} b - ω^{2} b) sin ω t + 2 ω (\partial_{t} b) cos ω t .$