An E8–Zeta Spectral Program for the Riemann Hypothesis: Exact Identities, Finite Weil-Form Diagnostics, and Obstructions

We examine whether the E8 lattice can contribute nontrivial arithmetic data to spectral programs for the Riemann Hypothesis (RH). This article does not prove RH, conditionally or unconditionally. The classical identity Z_E8(s) = 240·2⁻ˢ·ζ(s)·ζ(s−3) is used to derive an exact reconstruction formula for the von Mangoldt function from the logarithmic derivative of Z_E8, showing that the E8 Epstein zeta function contains the prime-power data entering the Riemann–Weil explicit formula. We also give a formal fermionic Euler-product cancellation of the shifted factor ζ(s−3). These facts do not imply RH: a positive self-adjoint gapped operator can have ζ(s) as its spectral zeta without constraining the zeros of its meromorphic continuation — a claim of the form "mass gap ⟹ RH for a spectral zeta" is false, as the elementary counterexample H|n⟩ = n|n⟩ shows. We audit finite Weil-form computations at fixed arithmetic cutoff and identify that the previously reported spectral gap (~5×10⁻³¹) is a likely numerical artifact, traced to an artificial cutoff of the archimedean integrand at x<10⁻³⁰ where the true integrand has a finite nonzero limit; the reported "certified" bounds discarded the quadrature routine's own error estimate and did not establish uniform positivity, only Cauchy-interlacing monotonicity, which is compatible with convergence to zero. The (E8)₁ affine character must not be confused with the E8 lattice theta series: the coefficients 240σ₃(n) count lattice vectors in shells and are not automatically the multiplicities of a physical current operator on AdS2 without an independently constructed Hilbert space, domain, and boundary conditions. The correct long-term objective is a Hilbert–Pólya construction: an independently defined self-adjoint operator D_E8 satisfying det_reg(z − D_E8) ∝ Ξ(½+iz), not a mass-gap argument applied to a spectral zeta function.