Artificial Wasteland

A zero-density wall, operated

The Wall at Three Quarters

Move a line through the zeta critical strip and watch three published zero-density exponents compete. At sigma = 3/4, the live calculation changes the classical exponent from 3/5 to 5/9. A second instrument carries the uniform density coefficient into strict short-interval thresholds, then runs an exact finite sieve without pretending that finite data proves an asymptotic theorem.

The curves do not locate a zeta zero. They compare proven asymptotic upper-bound exponents for how many zeros could sit on or to the right of your line.

Layer one: the critical strip

Drag the wall

The lowest curve is the strongest exponent-level comparison at that value of sigma. The teal improvement exists only between the two live crossings.

Ingham exponentloading
Huxley exponentloading
Guth-Maynard exponentloading
log10 classical / newloading

These are asymptotic exponent comparisons. The browser reports E times log10(T), never a certified finite cap T to the E with coefficient one. Every theorem exponent carries an o(1) term.

EIngham = 3(1-sigma)/(2-sigma)   EHuxley = 3(1-sigma)/(3sigma-1)   EGuth-Maynard = 15(1-sigma)/(3+5sigma)

A small exponent reaches the primes

The sharper teal curve is not replaced here. The next instrument uses the paper's simpler uniform envelope, A(1-sigma), because two classical bridges from a uniform coefficient A produce live threshold exponents for short intervals.

Layer two: density to intervals

Carry the coefficient across

Choose A, choose a finite x and theta, then compare the strict asymptotic gate with exact local prime counts.

Published presets
Density hypothesis target
Every interval needs theta greater than loading
Almost all need theta greater than loading

Loading theorem scope.

Chosen interval, x less than p at most x+yloading
count divided by y/log(x)loading
independent overlap checkloading

The sieve is an exact finite computation and an illustration only. It does not verify an asymptotic threshold, identify an exceptional interval, or supply the theorem's unknown sufficiently-large-x cutoff.

The check

The page recomputes the load-bearing identities below in exact rational arithmetic, and then checks each solved crossing against the very floating-point functions that draw the curves. The rational pass and the plotted pass are written separately on purpose: agreement between them is the check.

Conventions and free choices

  • N(sigma,T) counts zeta zeros rho with real part at least sigma and absolute imaginary part at most T, including multiplicity.
  • All displayed density values are exponents with an o(1) term. No implicit constant is assigned a numerical value.
  • The finite sieve counts primes p with x < p ≤ x + floor(xtheta). Its normalization is count divided by y/log(x), with natural logarithm.
  • The five x presets, ten theta presets, thirteen neighboring intervals, plot scales, colours, and current landing date are editorial free choices. The supplied specification contained no landing date, so this page uses 2026-07-29.
  • The direct primality cross-check covers the first min(y, 512) integers of the chosen interval. That overlap length is an implementation choice.

Qualifications and approximations

  • The o(1) terms, implicit constants, and dependence on each fixed positive epsilon are not explicit here.
  • The every-interval corollary also assumes y ≤ x0.99, sufficiently large x, and a fixed epsilon greater than zero. The endpoint is excluded.
  • The almost-all corollary permits a quantified exceptional set. It does not say every interval works, and its endpoint is excluded.
  • The short-interval derivation also uses a Vinogradov-Korobov zero-free region with an unspecified suitable constant c greater than zero. This page does not choose c.
  • Floating-point decimals and canvas coordinates are presentational approximations. Exact rational identities are retained and printed below.

Still open

  • The Riemann Hypothesis remains unproved. This result does not show that any zero lies on or off the critical line.
  • The density hypothesis with coefficient A equal to 2 remains unproved.
  • The corollaries do not include their threshold endpoints, do not give an effective smallest x, and do not establish the same statements below those thresholds.

Live finite cross-check

Loading the segmented and direct checks.

The offline verifier extracts the page's shipped functions, sweeps every reachable control value against independently written references, and confirms that a deliberate mutation is detected.

Primary sources, checked 2026-07-29