What if the constants of nature could be calculated?
Boundary Phase Resonance is an attempt to answer that question with a discrete model of space, two integers, and a public record of every result, including the ones that failed.
Start here Skip to the mathematicsThe question
Physics has a short list of numbers it cannot explain. The strength of electromagnetism is one of them: about 1/137. The masses of the electron, the muon, and the quarks are others. The Standard Model of particle physics needs roughly two dozen such numbers typed in by hand. It describes what happens with extraordinary accuracy, but it does not say why those particular numbers.
Most physicists treat this as a fact of life. A few frameworks, string theory among them, hope the numbers fall out of a deeper structure, but so far none has produced them. BPR is a much smaller and more concrete attempt at the same goal. It does not claim to be finished. It claims to be checkable.
The goal
BPR's ambition is a theory of everything: one structure from which the constants, the forces, and the particles all follow, with nothing typed in by hand. That is the goal, stated plainly, and it has not changed. What has changed is how much of it the project claims to have already reached. The first version claimed most of it. The current version claims one working sector, states the rest as open problems, and keeps a public record of every step that was checked and found wanting. The distance between the goal and the evidence is the honest measure of the project, and this site tries to show it exactly.
The picture
Imagine that space, at the smallest scale, is not a smooth fabric but a grid. Each point on the grid holds a single quantity: a phase, like the position of a clock hand. The clock has not infinitely many positions but a fixed, very large number of them. In BPR that number is a prime, written p. Each point is connected to a fixed number of neighbours, written z.
Now imagine the edge of a region of this grid. Phases on that boundary can line up with each other, or resonate, in patterns. BPR proposes that what we call particles, forces, and fields are the long-wavelength shadows of those boundary patterns. The framework's name comes from this: resonance of phases on a boundary.
Two things make this more than a metaphor. First, the boundary is taken to be a sphere, and there is an argument for why it must be a sphere rather than, say, a doughnut. Second, once p and z are fixed, the model produces numbers, and those numbers can be compared with experiment.
A prime number. It sets how many phase states each point can hold. It is fixed by asking the framework's own formula for electromagnetism to match the measured value, then choosing the nearest prime that satisfies an orientability condition (p ≡ 1 mod 4).
The number of neighbours per point. It is the coordination number of a cubic tiling on a two-sphere boundary.
Honest note: p is not derived from nothing. It is anchored to one measured constant, the fine-structure constant, and then checked against three other relations that point to the same prime. The details are on the how it works page.
One formula you can check tonight
The fine-structure constant, α, sets the strength of electromagnetism. BPR's expression for its inverse is
where γ ≈ 0.5772 is the Euler–Mascheroni constant. With p = 104,761 and z = 6 this gives 137.039. The measured value is 137.036. The difference is 0.002 percent.
You should be suspicious of this, and so are we. The prime was chosen to make this formula work, so the agreement is partly built in. What is not built in is the form of the formula, the fact that a prime near 105 is also singled out by three unrelated relations in the framework, and the fact that the same p and z then feed everything else. Whether that is a coincidence or a clue is the open question. The calculator lets you move p and z and watch what happens.
What it gets right
With the two integers fixed, the framework produces a long list of quantities. Some agree with experiment surprisingly well. A few examples, with the status the project assigns to each:
| Quantity | BPR | Measured | Status |
|---|---|---|---|
| Inverse fine-structure constant 1/α | 137.039 | 137.036 | derived p anchored to this |
| Solar neutrino mixing, sin²θ₁₂ | 0.3086 | 0.309 ± 0.009 (JUNO 2025) | framework one fitted coefficient |
| Reactor neutrino mixing, θ₁₃ | 8.64° | 8.54° ± 0.15° | derived |
| CKM CP phase, δ | 68.5° | 68.5° ± 5.7° | derived |
| Muon to electron mass ratio | 210 | 206.77 | conjectural 1.5% off |
| Sum of neutrino masses | 0.060 eV | < 0.12 eV | derived |
The full scorecard, with every status label explained, is on the results page. Read the labels. "Derived" means the number follows from p and z with no adjustment. "Framework" means the formula is BPR's but at least one coefficient was fitted. "Conjectural" means the formula reproduces a number but no derivation from the model's equations exists yet.
Where it has been set back
This is the part most theory websites leave out, and the part we think matters most. None of it means the goal was wrong. It means the first version claimed it too early.
The "theory of everything" claim was withdrawn in 2026. Withdrawn, not abandoned: the goal stands, the claim to have reached it does not. The original version of BPR claimed to contain the strong nuclear force. A blind test was set up: the masses of certain bound states of the strong force, called glueballs, were locked in before any calculation, and the model was then asked to reproduce their ratios. It could not. The boundary field in BPR 1.0 provably cannot host a light parity-odd state, and its predicted mass ratio fell outside the pass band. So the 1.0 particle sector, as built, cannot be the whole story. The headline claim was set aside, a replacement sector was proposed, and the project went back to work. A theory that honours its own kill conditions has a better chance of eventually earning the claim than one that argues around them.
Two foundational derivations were withdrawn in September 2026. A step that claimed to derive the number three, for the three families of matter particles, from the geometry of a sphere turned out to be invalid. So did a topological argument behind one of the quark mass formulas. The numbers those formulas produce are unchanged, but they are now labelled as conjectures, and the family count is treated as an input from experiment, not a prediction. The corrected mathematics is on the mathematics page.
Everything the project has determined not to work is kept in one place, the negative-findings registry. The status page tells the story in order.
Where it stands today
BPR 2.0 is best described as an exploratory framework with one working sector and one open problem.
- The working sector is flavor: formulas for mixing angles, some mass ratios, and neutrino properties that agree with data and, in a few cases, were registered before the measurements came in.
- The open problem is the particle sector. A proposed replacement, built from a gauged discrete symmetry of the lattice, gives a legitimate two-dimensional theory of exotic quasiparticles called anyons. It does not yet give the three-dimensional strong force, the chiral matter of the Standard Model, or a particle spectrum. Nobody should read the proposal as more than a proposal.
Whether BPR is a collection of striking coincidences or the edge of something real will be settled by experiments and by the open problems above, not by argument. The experiments page lists the measurements that would rule it out and when they are expected.
Go deeper
How it works
The four steps from a lattice to a measurable number, in plain language with the key equations alongside.
Mathematics
The action, the field equation, the spectrum on the sphere, the flavor ansatz with its honest labels, and the two withdrawn proofs with their corrections.
Results
Every benchmarked quantity with its status and its distance from experiment.
Experiments
What would falsify BPR, which instrument can do it, and when.
Status
The timeline: the 1.0 claim, the blind glueball benchmark, the 2.0 narrowing, and the 2026 foundation repair.
Run it
Clone the repository, run 1,600 tests in about a minute, and reproduce every number on this site.
Questions people ask
Has this been peer reviewed?
No. The paper, the code, and the full record of failures are public so that anyone can check the work now. Formal review has not happened, and nothing here should be treated as established physics.
Isn't picking p to match α just cheating?
Choosing p from α makes the α agreement unsurprising, and the site says so wherever it appears. The case for taking p seriously rests on everything else it feeds. Three other structural relations in the framework each point to a prime within a few percent of the same value. That could still be coincidence. It is the kind of coincidence worth testing.
Why a sphere?
The argument is that a consistent boundary must be compact, orientable, and simply connected, and the only closed surface with all three properties is the two-sphere. That part is sound mathematics. What was withdrawn is the further claim that the sphere's geometry forces exactly three families of matter. It does not. Three is an input.
What happened to the "theory of everything" claim?
It is still the goal. In 2026 a blind test showed that the first version's particle sector could not reproduce the strong force, so the claim to have already reached the goal was withdrawn. The framework was narrowed to what it can currently support and a new particle sector was proposed. Earning the claim back means, at minimum, a particle-physics bridge that gives the strong force and chiral matter, a pass on the sealed benchmark that the first version failed, and at least one fermion mass label derived rather than fitted. The status page tracks each of these.
What would prove it wrong?
Several things, each tied to a scheduled experiment. If the JUNO observatory finds the neutrino masses are in inverted order, the boundary topology argument fails. If neutrinoless double beta decay is observed, the orientability condition fails. If precision Casimir force measurements find an exponent far from 2, the projection postulate fails. The experiments page has the full list with thresholds.
How is this different from string theory or loop quantum gravity?
It is far less ambitious and far more specific. BPR works in ordinary 3+1 dimensions, has two integer inputs, and produces particular numbers that can be compared with today's data. It also currently lacks what those programs have: a consistent account of gravity and the strong force at the quantum level. It should be judged as a source of testable coincidences, not as a rival to those programs.
Can I run it myself?
Yes. The repository is MIT licensed. See run it.