The primes have a spectrum.

The prime numbers look irregular. They are not. Every jump in the prime staircase can be rebuilt from a list of numbers that says nothing about primes at all — the zeros of the Riemann zeta function. Below, you can add those zeros one at a time and watch the staircase assemble itself.

THE FORMULA, ENTIRE

ψ(x) = x − Σ_ρ x^ρ/ρ − ln(2π) − ½·ln(1 − x⁻²)

Read it right to left. The last two terms are constants. The first term, x, is the smooth trend. Everything interesting is the middle sum, taken over ρ = ½ + iγ, the nontrivial zeros. Each zero contributes one wave. Add enough waves and you get the exact position of every prime.

FIGURE 01 · HARMONIC BUILD-UP

Add zeros. Watch the staircase appear.

ZEROS 100WINDOW 2– 200— ψ(x) by sieve · — reconstruction from 100 zeros

FIGURE 02 · ONE INSTRUMENT

What a single zero sounds like

ZERO # 1γ = 14.134725period in ln x = 0.4445

FIGURE 03 · A CORRECTION WE FOUND MID-SESSION

ψ does not go flat in a prime gap

THE INTERVAL (113, 127) · 500 ZEROS — ψ jumps by ln 11 at 121 = 11^2 · ψ jumps by ln 5 at 125 = 5^3

TABLE 01 · LOCKED BASELINE

The gate any change has to pass

ZEROS	LOCKED MAX \|ERR\|	LOCKED MEAN	YOUR BROWSER · MAX	YOUR BROWSER · MEAN
5	11.00	2.38	—	—
100	5.85	1.33	—	—
500	3.34	0.65	—	—

Evaluated on half-integers in [2.5, 999.5] — half-integers because they sit away from the jumps, where the value is unambiguous. The live columns are computed in your browser against the same sieve; they should match the locked columns exactly.

At zero zeros the reconstruction is a straight line — the smooth trend, no information about individual primes. At five, broad waves appear in roughly the right places. By a hundred the steps are legible. By five hundred the reconstruction sits on the staircase, and the remaining error is concentrated exactly where theory says it must be: at the jumps themselves.

THE CORRECTION, IN FULL

There are no primes between 113 and 127. An early version of our validation gate checked that the reconstruction went flat across that interval, and treated the wobble it found as Gibbs ringing — an artefact. It was not an artefact. ψ counts prime powers, not primes. Inside that gap sit 121 = 11² and 125 = 5³, and ψ jumps by ln 11 and ln 5 at exactly those points. The 500-zero reconstruction was tracking both — correctly. The test was wrong, not the engine.

Any claim of the form “flat inside prime gaps” has to read “flat between prime powers.” It is a small correction with a sharp lesson: the apparent noise was signal, and only a falsification test found it.

RULE THAT CAUGHT IT — Gate before build. Any new claim gets a cheap falsification test before further construction is allowed on top of it.

PIPELINE A

The sieve

A von Mangoldt sieve gives ψ(n) exactly, by counting. It knows everything about primes and nothing about zeta zeros.

PIPELINE B

The reconstruction

The explicit formula rebuilds ψ from zero ordinates alone. It never calls a primality test — that is the rule that makes the result mean anything.

ORDERING

Ascending, conjugates paired

The zero sum is conditionally convergent, so reordering it is invalid by Riemann rearrangement. Zeros are consumed in ascending order, always.

NO ORACLES

No snapping to primes

Any construction whose final step rounds to the nearest prime is invalid by design — the oracle did the work, not the formula.

ZERO ORDINATES: A. M. ODLYZKO'S TABLES, FIRST 500, 9 DECIMALS · ENGINE RUNS ENTIRELY IN THIS PAGE