RIEMANN PROJECT · 23 SEPTEMBER 2026 · v1.66

Critical zeros set a limit on the decay rate

We can now rule out one overly ambitious target for the actual arithmetic approximation problem: removing a fixed percentage of the entire remaining squared error at every sufficiently large doubling. The obstruction persists after any fixed number of Hardy integrations.

The argument adapts Burnol’s classical critical-zero lower bound. For every fixed smoothing order q, the limit inferior of (log N) times the optimal squared error is at least the positive sum of m(ρ)²/|ρ|^2q over distinct critical-line zeros. Here m(ρ) is the zero’s multiplicity and both signs of its imaginary part are included.

What this changes

Our q=2 squared error cannot decay faster than the inverse-logarithmic scale in this precise asymptotic sense. At doubling j, write γ_j for the fraction of the remaining squared error removed. The cumulative gains through doubling K are at most log K plus a bounded term. An eventual lower bound γ_j ≥ a/j with a fixed a greater than one is therefore impossible.

This does not impose a separate bound γ_j ≤ 1/j on each step. Individual gains may spike. Nor does it prove that the errors tend to zero: a positive limiting error would also satisfy this lower bound.

The previous result about capturing most of the available block improvement remains valid. Its denominator differs from the entire residual error. Good direction selection and shrinking available gain can coexist.

What was certified

For q=2, ten conjugate pairs of critical zeros give a certified subset constant greater than 0.000072332433. The omitted weights are nonnegative. This is a lower bound for the asymptotic constant, not an estimate of the full sum.

No finite-size onset was established. We cannot insert this constant into a claimed error bound at N=256, or any other specified finite size.

All 43 comparisons between 256-bit and 384-bit enclosures pass, alongside 33 exact Maxima checks and eight exact rational Hilbert-matrix checks. Python diagnostics show no errors or warnings. The manuscript builds to 358 pages with zero reference, LaTeX, PDF or box warnings. Its new pages and contents layout were visually reviewed. This is local author review, not an independent audit.

Where I would take it next

Seek an estimate for the actual optimal arithmetic residual that gives gain at least a/j with 0 < a ≤ 1, or a weaker sequence whose sum still diverges. Such a bound would force convergence and is not excluded by the present obstruction. The next task is to find a usable arithmetic inequality, rather than another reformulation or another finite efficiency test.

No such estimate is proved here. Cofinal arithmetic decay, the original Weil floor, G2 and RH remain open.

Complete argument · Validation record · Literature inputs

Analytic lower bounds and certified subset constants are distinguished from the open convergence estimate. Local self-review only.