The pole term is the only obstruction to Perron structure in the localized Weil quadratic form: a rank-two splitting and a scalar criterion for the bottom eigenvalue

Let A_a be the self-adjoint operator on L^2(-a,a) associated with the localized Weil quadratic form of Connes-Consani-Moscovici (arXiv:2511.22755) and Suzuki (arXiv:2606.09096), realized through the screw function g of the Riemann zeta function. The missing analytic step in the CCM strategy toward RH, as stated by Connes (arXiv:2602.04022, Section 6.6), is that the smallest eigenvalue of the (truncated) Weil form be simple with even eigenfunction. Suzuki proved this for the continuum operator for sufficiently small a; we locate the obstruction to extending the classical positivity argument: the smooth off-diagonal kernel of A_a is -g''(t) = 2cosh(t/2) - e^{-t/2}/(1-e^{-2t}), which violates the Beurling-Deny sign condition precisely for |t| > t* = 0.28119957..., confining the classical route to a <= t*/2 ~ 0.1406.

The main observation is that the violating term 2cosh((x-y)/2) is exactly the pole term W_{0,2} of the explicit formula - an explicit rank-two operator (kernel identity verified to 1e-301) - which splits by parity as +2|C><C| on the even sector (C(x)=cosh(x/2)>0) and -2|S><S| on the odd sector. Consequently the pole-free part (the archimedean-plus-primes part of the Weil form) has nonpositive off-diagonal kernel for ALL a>0: the sign barrier was entirely produced by the poles of zeta. An explicit form identity (verified to machine precision) exhibits the pole-free part as a positive jump Dirichlet form plus a lower-bounded potential; by Beurling-Deny, irreducibility and Perron-Frobenius this yields, with NO restriction on a, a simple, strictly positive and even ground state - removing the "sufficiently small a" restriction of Suzuki.

Since the pole term is rank one per parity sector, the even-sector spectral problem reduces to a one-dimensional Krein analysis: even-simplicity of A_a is equivalent to the pointwise positivity of the single explicit function (A~^even - lambda_0)^{-1}C. High-precision computations validated against defining series (cf. DOI 10.5281/zenodo.20671635) verify the full mechanism at c=53. The general-a positivity remains open; we reduce it to a renormalized nodal identity for the screw-function/Krein-string realization under a nonlocal rank-one perturbation, and isolate why no existing oscillation theorem (Gesztesy-Simon-Teschl; Remling-Scarbrough; Kruger-Teschl) yields it directly, leaving that lemma to a sequel.

Contributions: (1) the rank-two splitting identifying the poles of zeta as the sole obstruction; (2) the unconditional Perron theorem for the pole-free part (all a); (3) the exact reduction of the CCM even-simplicity hypothesis to one scalar-resolvent/nodal question. No claim about the Riemann Hypothesis is made.