A Theta–Regularized Identity for SL2 and a Fejér–Windowed Strip Bridge for Log|ξ|

A theta--regularized inner product identity in rank one is established, linking a mixed theta--weighted Eisenstein pairing on \( \Gamma \)\H to the \( \sigma \)--derivative of \( \log|\xi(s)| \), up to explicit Euler factor correction terms arising from the \( G\times G \) doubling formalism. More precisely, for \( s=\tfrac12+\sigma+t \) it is shown that \( \frac{\partial}{\partial\sigma}\log\left|\big\langle\Theta(\cdot)E(\cdot,s),\ \Theta(\cdot)E(\cdot,1-\overline{s})\big\rangle_{\mathrm{reg}}\right|=2\,\mathrm{Re}\,\frac{\xi'(s)}{\xi(s)}\ -\ 2\,\mathrm{Re}\,\frac{\zeta'(2s)}{\zeta(2s)}\ +\ 2\,\mathrm{Re}\,\frac{\zeta'(2-2\overline{s})}{\zeta(2-2\overline{s})} \),as an identity of tempered distributions in t. On the critical line \( \sigma=0 \) the Euler corrections cancel and a particularly simple formula is obtained:\( \frac{\partial}{\partial\sigma}\log\big\langle\Theta(\cdot)E(\cdot,\tfrac12+\sigma+t),\ \Theta(\cdot)E(\cdot,\tfrac12-\sigma+ t)\big\rangle_{\mathrm{reg}}\Big|_{\sigma=0}=2\,\frac{\partial}{\partial\sigma}\log\left|\xi\bigl(\tfrac12+\sigma+t\bigr)\right|\Big|_{\sigma=0} \). Fejér--windowed versions of these identities are then obtained, and a Fejér--windowed "strip bridge'' is proved: a harmonic operator identity expressing the short--band component of \( \partial_\sigma\log|\xi(1/2+\sigma+t)| \) at an interior latitude via a linear combination of Fejér--smeared edge data, with a power--saving \( O(H^{-\eta}) \) remainder after short--band freezing, uniformly for \( |\sigma^\star|\ge \sigma_0>0 \). A sharp truncation stability result is also established. After subtracting the finitely many Zagier--Arthur cusp counterterms, the Fejér--smeared \( \sigma \)--derivative of the logarithm of the truncated mixed theta--Eisenstein pairing agrees with its regularized version up to \( O(H^{-A}) \) for any prescribed \( A>0 \), provided the truncation height \( Y=H^{B(A)} \) is chosen sufficiently large. A brief discussion is included of numerical checks in a sample region, and a short Fourier--analytic proof note is given for the renormalization estimate that underlies the strip bridge.