Balance Formula Lab
The whole system in one formula, driven by two sliders. Send a test wave into the system: a window in log-time, centred on a height t₀, with a chosen sharpness. Three things happen at once. The prime powers answer with echoes at delays log n (each weighted log p / √n), the mirror at the corner and the sphere factor supply the input energy, and the drum's standing waves — one per zero — respond with a squared, never-negative amount. Riemann's explicit formula says: input − echo = zero side, exactly, for every window. The Riemann hypothesis is the statement that the right-hand side can never be negative for any window — that the echoes never out-shout the mirror.
The test wave exact — both sides computed independently
Prime side — echoes at delays log n, caught by the window
Zero side — the squared window over the zero ladder
The drum's response — the standing wave the window excites (its height at the centre is the zero side)
Zero side ÷ input as the window slides — the echo reinforcing on zeros, cancelling in gaps
Prime side (top left). Every prime power n is a stem at delay u = log n with height log p/√n, mirrored at −log n by the fold. The test wave h(u) = e−u²/2s² cos(t₀u) is drawn over them; the echo is the sum of stem × wave at each stem — green where a prime adds to the echo, amber where it subtracts. Zero side (top right). The window's squared response ĥ(t) is a bump at ±t₀ of width about 1/s. Each zero under the bump contributes ĥ(γ), drawn as a dot; the zero side is their sum, and it is a sum of squares. The balance. The bars show input (pole + sphere terms), echo, and the zero side; the small number is the discrepancy between the two independent computations, which stays near zero. The response. The standing wave the window excites in the drum, Σγ ĥ(γ) cos(γu); its value at the centre is the zero side. The ratio. Sliding t₀, the zero side divided by the input. Above 1 the echo is negative: the primes' echoes return with the mirror, which happens when the window sits on zeros. Below 1 the echo is positive and subtracts, which happens in the gaps, and there it falls toward zero — the knife edge — and falls further as the window sharpens. So the primes' echo flips sign between zeros and gaps; that is the explicit formula read from the primes' side. A proof has to show the echo never overshoots the input, in any gap, for any window — including the ones nobody will compute.