Focusing the direct route on CMP and ES

STATUS: RIEMANN HYPOTHESIS OPEN.

Public update: 2026-10-01. Research scope: completed audits and comparisons dated 2026-09-30. This page reports those results; no new proof or numerical experiment was performed for publication.

What changed

Reviewing the existing work showed that much of the general machinery—finite test functions and operator theory on fixed domains, for example—can be delegated to established theorems. The selected direct route now concentrates on two unproved obligations: the actual finite Weil ground must eventually be a unique even state, and its transform must approach a known prolate approximation sufficiently fast throughout the required complex domain.

The first question is ES; the second is CMP. Their scope is now more concrete: parity ordering and an internal spectral gap for ES, and an explicit sufficient convergence rate for CMP. Neither has been proved.

Why this direction

Rather than rebuild general results already available in the literature, the review isolates the estimates specific to the actual zeta system. Two central obligations remain on this chosen route; this is not a claim that RH is nearly proved or that its remaining difficulty has been counted.

Current boundary

Ordinary $$L^2$$ convergence is insufficient for CMP. The finite simple-even certificate at one parameter point does not establish eventual ES. Both obligations must hold on the same admissible cofinal sequence: one sequence along which the cutoffs grow while the necessary resolution conditions remain valid.

Next action

Investigate independently why the actual ground approaches the proxy at the required rate and why the ground eventually becomes simple and even. Keep the scope on CMP and ES; do not automatically add a third route, a new truncation, or a reconstruction of general operator theory.

Technical details

What the theorem-import count means

CASE D — CMP OPEN + ES OPEN. Of 19 items reviewed, 12 can delegate general theory or fixed-scope proofs to existing mathematics. This does not mean that 12 RH obstacles were solved. Newly discharged asymptotic estimates for the parameter-dependent actual Weil system: 0.

CMP: a value-normalized complex comparison

Let $v_{\lambda,N}$ be the actual finite Weil ground and $k_\lambda$ the known prolate proxy before head projection. With $$z_*=i/4$$ , CMP requires, on one cofinal sequence, for every compact set $K\Subset\Omega$ ,

\Omega=\{z:|\Im z|<1/2\},\qquad \sup_{z\in K}\left| \frac{\widehat v_{\lambda,N}(z)}{\widehat v_{\lambda,N}(z_*)} -\frac{\widehat k_\lambda(z)}{\widehat k_\lambda(z_*)} \right|\longrightarrow0.

The normalization denominators must be nonzero. Proxy convergence to $\Xi$ is known in its stated scope; comparison with the actual ground is not.

One sufficient condition, called CMP-R, uses $a=\log\lambda$ , support restriction $$R_a$$ to $$[-a,a]$$ , finite Fourier projection $$P_N$$ , and orthogonal projection $$P_v$$ onto the line spanned by the chosen unit ground:

D_{\lambda,N}=\|(I-P_v)P_NR_a k_\lambda\|_2,\qquad \lambda^rD_{\lambda,N}\longrightarrow0 \quad(0<r<1/2).

This limit must hold for every fixed $$0<r<1/2$$ . The head-resolution condition $\lambda^r a/(N+1)\to0$ must also hold for each such $$r$$ on the same sequence. The actual comparison estimate is unproved. This is a sufficient condition, not an asserted equivalence with CMP. Expanding support increases the norm of complex evaluation, so ordinary $$L^2$$ convergence alone does not suffice.

ES: eventual simplicity and evenness

Write $$e_+(a,N)$$ and $$e_-(a,N)$$ for the lowest even- and odd-sector eigenvalues, and $\Delta_+(a,N)$ for the gap between the lowest and next even-sector eigenvalues. The target is that eventually

e_-(a,N)>e_+(a,N),\qquad \Delta_+(a,N)>0

on the same sequence used for CMP. These conditions give a simple even ground for the full finite system.

At the single endpoint $c=\lambda^2=13$ , $$N=4$$ , the certification inheriting the saved interval enclosures confirms a simple even ground. It is not generalized to the infinite limit or eventual ES. The unproved CMP comparison is not used to justify ES.

What can be imported afterwards

If CMP and ES hold together and all hypotheses are verified, the finite real-zero theorem, prolate-proxy convergence to $\Xi$ , and Hurwitz/Rouché implications can be supplied by existing results. That application has not been completed.

EVEN FULL-GROUND CAPTURE: NOT ESTABLISHED.

Evidence and sources

Recorded editorial research state · Current state · Roadmap · Earlier history · Finite certificate and its limits

The publication records the creator-confirmed summary of completed work. Existing source records and evidence hashes are retained; the count of imported steps is not new mathematical evidence.