18. Cyclicity of the even derivative family — dated update

STATUS: RIEMANN HYPOTHESIS OPEN

Update recorded 2026-09-30, after the 17-track publication inventory. This update completes Priority 1 only. “Full ground capture” is the source directory name, not a theorem proved in this update.

Starting idea

The fixed finite-dimensional selection theorem leaves a basic question: does the union of the tested derivative families even reach every even square-integrable function? Density is a necessary approximation resource, but supplies no uniform rate for a growing family.

What was tested

Use the actual normalization

\widehat{k}(x)=\int_{\mathbb R}k(t)e^{-ixt} dt=\Xi(x)/4, \qquad \Xi(x)=\xi(1/2+ix).

The proof studies the positive measure $d\mu(x)=|\Xi(x)|^2dx$ . Standard Gamma decay and an unconditional zeta bound imply a finite exponential moment for every exponent below $\pi/2$ . Since Xi is a nonzero entire function, its real zeros have measure zero; RH and simplicity of those zeros are unnecessary.

What survived

The unconditional conclusion is

\boxed{\overline{\mathrm{span} _{\mathbb C}\lbrace k^{(2j)}:j\ge0\rbrace}^{ L^2(\mathbb R)} =L^2_{\mathrm{even}}(\mathbb R).}

The real-span version also holds in real even L². Carleman's condition gives polynomial density in L²(μ); even symmetrization, the onto isometry $h\mapsto\Xi h$ , and Plancherel give the result. A second proof uses analytic Fourier transforms and uniqueness. Neither proof divides by a uniformly bounded inverse of Xi.

For the change of variable $$y=x^2$$ , the Stieltjes moment condition uses $m_{2n}^{-1/(2n)}$ . It must not be replaced by the different Hamburger condition involving $m_{4n}$ . A supplementary moment asymptotic is not needed for the density proof.

What remains unproved

L² density does not establish density in the Weil form domain, a form core, a uniform approximation bound, stable coefficients, or a gap outside a fixed derivative family. It does not justify exchanging the limits in support, Fourier resolution and derivative order. The known radical identities cannot be extended using unproved continuity or closability in L².

Full ground capture, estimates uniform in growing m, the ES ground-state hypotheses, G*, and RH remain open. Priority 2 and later tasks were not undertaken in this update.

Why the work stopped here

The requested cyclicity question has a positive, unconditional answer. A new argument would be needed to convert qualitative density into the quantitative and form-topological control required for ground selection. This page records that boundary rather than treating density as completion of the larger program.

Current scope

Cyclicity: proved in the written argument, with scoped internal AI-agent audit. The proof applies established moment and Fourier principles; no novelty claim is made. It has not been externally peer reviewed or formalized in Lean.

Canonical records

Research timeline · Reproducibility · References

A later, separately scoped Priority 2 update addresses fixed finite-head rank and conditioning. It does not change what was established at the time of this cyclicity update.