CRITICAL-HALF: Green/tree/scattering bridge and shorted NP prolate front

[Waschtl904]

Scope

Research branch after NULLPOL-COMMON. Adds theorem-level structural audits for the critical-half Green/tree/Poisson/scattering architecture and reduces the remaining fixed-window NP-GAP problem to a rank-2-shorted weighted prolate/Birman–Schwinger certificate.

Key exact results include:

• NULLPOL = Range(-d^2+1/4) on compact windows;
• P11 AR(1)/hub-rest as the sampled critical-half Green/OU tree;
• pole transfer, Gamma resolvent ladder, finite-part diagonal mass, Poisson/scattering phase;
• pointwise scattering-positivity no-go;
• monotonicity of the centered gap under window exhaustion;
• exact rank-2 Paley–Wiener shorting and trace-class negative block;
• trace-minus-Ritz certification lemma for a rigorous a=1/2 prolate test.

Nonclaims

• no all-window NP-GAP proof;
• no RH proof;
• no Registry promotion;
• floating/Nyström margins in the latest audits are diagnostics only until Arb-certified;
• publication novelty remains open.

Keep as Draft while the a=1/2 Arb certificate is built and reviewed.