Golden Physics Project · Verified mathematics
Physicists who study how particles scatter across the sky found a symmetry they call the shadow. Mathematicians have had the same symmetry for more than 160 years, in Riemann's work on the prime numbers. The Golden Physics Project treats the two as one mirror, and asks how much each world can explain about the other.
This site is where that is checked, by a computer, one logical step at a time, with nothing taken on trust. It also says plainly what has not been shown.
The idea
Picture a plane of numbers with a gold line running through it. Every point has a partner on the other side of the line, at the same height and the same distance away. Points on the line are their own partners.
On the physics side, this reflection is the shadow map, which pairs each way a particle can spread across the celestial sphere with a partner. On the mathematics side, it is the symmetry of Riemann's zeta function, the function that encodes the prime numbers.
Riemann's zeta function has special points called zeros, and they always come in mirror pairs. Every zero found so far, trillions of them, sits exactly on the gold line. The Riemann Hypothesis says they all do. It has been open since 1859.
Tap or click the plane to place a point and see its mirror. With the keyboard, focus the plane and use the arrow keys.
Why "holography"
A hologram is a flat film that stores a whole three-dimensional scene. Mathematics has exact versions of the same trick: two descriptions of one thing, where everything on one side is encoded on the other. Here the lens between them is the Mellin transform, which turns sizes into frequencies. The framework finds the mirror at three nested levels.
Take all positive sizes and fold them at 1, with big on one side and small on the other. Seen through the lens, this becomes a sphere of frequencies cut in half by the gold line. Folding at 1 turns into the mirror across the line.
Boundary: sizes, folded at 1 · Hologram: the two halves of the sphere and the line between them
The whole numbers behave like the harmonics of a vibrating string, and that pattern looks the same whether you zoom in or out. Through the lens, that sameness is exactly Riemann's mirror symmetry of the zeta function.
Boundary: harmonics of a circle · Hologram: the functional equation ξ(s) = ξ(1−s)
Every whole number is built from primes in exactly one way, so think of each prime as its own dial. Walking up the gold line turns every dial at once, at speeds that never fall into step. The walk passes arbitrarily close to every setting, so the line is written densely into the dials: a hologram inside a hologram.
Boundary: one dial per prime · Hologram: the gold line, wound densely through them
Where the Riemann Hypothesis sits
In hologram language, the mirror symmetry says the picture is symmetric. That part is known (Riemann proved it), and it is checked here.
The Riemann Hypothesis asks for more: that the picture is also positive, like a physical system with no negative energies. Every zero on the gold line behaves like a stable, positive mode. A zero off the line would come with its mirror twin, and the pair together always contains one negative "ghost" direction. You can see this in the mirror demo above.
The finite version of that statement is proved here exactly: positivity holds if and only if there are no mirror pairs. What nobody has done, here or anywhere, is prove the positivity for the real primes. That step is equivalent to the Riemann Hypothesis itself, and this project does not claim it.
What's been checked, and by whom
Every result on this site is written in Lean, a proof checker. Lean accepts a proof only if every step follows from the rules of logic. It doesn't care who wrote the proof or how convincing it sounds.
Lean is built by the Lean Focused Research Organization. Our proofs lean on Mathlib, a shared library written by hundreds of mathematicians. Neither was written by this project.
Every proof uses only Lean's three standard logical axioms, the same ones all of Mathlib uses. Nothing of our own is assumed, and an audit script checks this.
Every change is rebuilt from scratch on GitHub's servers, and the full history of those runs is public. Anyone can rerun it on their own computer in four commands.
sorry is Lean's word for "trust me, skip this step". There
are none. Open stubs are results we want but have not proved. Each is named
open_… and asserts nothing, so it can never be mistaken for a proof. CI rejects
any that are named otherwise, and it fails the build if a number on this page stops matching
the code.
What "checked" does and does not mean. A checked theorem is true given its stated assumptions. Where a result depends on physics (for example, a value from quantum field theory), that input appears as an explicit assumption in the statement, never as a hidden axiom. The link between celestial physics and the zeta function is an exact structural match. It is not an experimental result, and nothing here claims that the zeta function is literally a physical quantity.
How the pieces depend on each other
Read each column from top to bottom. Each step builds on the ideas above it. Green steps are proved and checked, and red steps are open. The exact Lean dependencies can branch, and they are all drawn out in the interactive dependency graph, which covers every result and every arrow.
A representative selection
Each card gives a plain summary, the exact Lean name, and a link to the source. A script checks every name, file and badge on this page against the code on every build. The blueprint covers everything else.
Don't take our word for it
With Lean installed, these four commands download the project and re-check every proof on your own machine. A green build means every stated theorem follows from its stated assumptions. It does not mean the framework as a whole is established.
The ideas come from Daniel Toupin's On the Nature of Nature. Exploratory numerics live in GPPDiscovery, where nothing counts as proved until it arrives here.