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RH Frontier: Session Summary (March 26, 2026)

Status: 25% confidence. Gap precisely located.

The Chain

theta(it) -> Phi(u)>0 [proved] -> xi(s) -> A(z)=Xi(z)/2, B=HT(A), E=A-iB. E in HB <=> RH.

Proved This Session

  • Phi(t)>0 all t: PROVED (omega(x)=x^{-1/2}omega(1/x), modular)
  • xi = A-component of HB, on boundary (|E|=|E*|): PROVED
  • B<0 at all numerical A-zeros (interlacing): CONFIRMED
  • L1(x)=(H')^2-H*H''>0: CONFIRMED numerically

Killed This Session

  • Bounded Lyapunov K (B*K+KB=0): strongly stable LP => K=0. Permanent.
  • Phi>=0 => HB: FALSE (counterexample: 1+2e^iz+3e^{2iz} positive kernel, zeros in UHP)
  • GL exact equality =_E: NUMERICAL KILL (Mellin weights differ: |t|^2 vs |2xi/s(s-1)|^2)
  • T_new=L^{-1/2}Psi isometric: requires L=Psi*Psi (polar decomp). Extra structure = RH.
  • Connes W>=0 from adelic geometry: he REDUCES RH to W>=0. Circular.
  • PF_5 total-positivity: Michalowski arXiv:2602.20313 certified failure at order 5.

The Gap — Precisely Located

Every bridge from physical/geometric positivity to analytic positivity hits:

GL L>=0, Phi>0, theta PD, adelic structure, GUE
        |
        |  THE GAP
        |
H(E) >= 0  =  zeros of xi all real  =  RH

Not technical. Structural. Every bridge either assumes zeros on line (circular) or requires inequality itself equivalent to RH.

Real Open Problems (Csordas Hierarchy)

  • L1(x)>=0: numerically true, analytically open, independent of RH
  • B/A Herglotz: blocked at PF_5
  • K_n positive definite for all n: open, equivalent to RH

GL->de Branges Architecture (Sound, Incomplete)

Psi = M o Pi_arith: H_fluct -> H(E) Psi_hat(s) = [2xi(s)/s(s-1)] * psi_hat(s) - Bounded: YES - Muntz density: YES (sum 1/log n = inf) - Bounded below: requires L=PsiPsi. Not free. Circular.

Core Insight

The difficulty lives in one step: physical positivity -> analytic positivity. Geometry (GL, adelic, modular) reaches the boundary and stops. Nash equilibrium is unique on Re(s)=1/2 — we see it, cannot yet certify it.