An experiment in AI-assisted theoretical physics
A note on how this was made
This project began as a question: what happens if you point a generative AI at one of the hardest unsolved problems in fundamental physics?
I’m not a professional physicist. I have a background in software and mathematics, and a long-standing obsession with the question of why the Standard Model has the structure it does — why three generations, why these masses, why these mixing angles. These are questions that decades of conventional research have failed to answer.
Over several hours, I worked with GitHub Copilot in an unusual collaboration.
I brought the original, overambitious guiding question:
ok, let’s think about some theories for finding a theory of everything that unites quantum mechanics and particle physics. We need an approach that tries some new marths, new ideas, and is based on all current observations and data. Make a plan (PLAN.MD) on how we’re going to solve this.
The project has since been deliberately down-scoped: the current claim is a Standard Model parameter framework, not a completed theory of everything.
The AI brought the ability to rapidly compute, cross-check against experimental data, explore algebraic identities, spot errors in derivations, and maintain consistency across a growing web of interconnected predictions.
The result surprised me. What started as exploratory numerology gradually locked into a rigid few-input structure with 23 grouped quantitative relations — all matching known experimental values to 0.1–7%. These are mostly postdictions, not historical predictions: the constants were already known. The AI didn’t hallucinate these agreements; it computed them from explicit algebraic formulas and checked them against published PDG and NuFit data. When a relation was wrong, we found out immediately and either fixed the derivation or acknowledged the discrepancy.
What the AI contributed: Rapid symbolic computation, error-checking, consistency enforcement across the code and notes, literature cross-referencing, and the sheer throughput to explore hundreds of algebraic possibilities in the time it would take a human to check one.
Is this real physics? That’s for the community to judge. The numbers are intriguing, but the continuum/RG derivation and several algebra-to-physics bridge rules still need hardening. The framework is falsifiable (several experiments in the next decade will test parts of it), and the markdown audit documents lay out what is derived, what is assumed, and what remains open.
Either way, it’s a story about what becomes possible when you combine human intuition with AI capability.
What if many constants of the Standard Model came from a single piece of mathematics?
The Triumph of the Standard Model
The Standard Model of particle physics is one of humanity’s greatest intellectual achievements. It predicted the existence of the Higgs boson decades before its discovery at CERN in 2012. It calculates the magnetic moment of the electron to 12 decimal places — the most precise prediction in all of science. It has survived every experimental test thrown at it for fifty years.
The Standard Model describes 17 fundamental particles interacting via three forces (electromagnetism, the weak force, and the strong force), governed by the mathematical structure known as the gauge group SU(3)×SU(2)×U(1). It is, by any reasonable measure, spectacularly successful.
And yet it raises a question it cannot answer: why this structure?
The theory contains 19 numerical parameters — particle masses, coupling strengths, mixing angles — that must be measured experimentally. It tells you how to use these numbers, but not where they come from. The mass of the electron, the strength of electromagnetism, the number of generations — all are taken as given.
This isn’t a failure of the Standard Model. It’s an invitation. What if there exists a deeper mathematical structure from which all 19 parameters can be derived?
Four Number Systems, and That’s All There Is
Mathematics provides a surprising constraint. There are exactly four division algebras — number systems where you can add, subtract, multiply, and divide without any operation giving zero from nonzero inputs:
| Algebra | Dimensions | Key property lost | Role in physics |
|---|---|---|---|
| Real numbers ℝ | 1 | — | Classical mechanics |
| Complex numbers ℂ | 2 | Ordering | Quantum mechanics |
| Quaternions ℍ | 4 | Commutativity (a×b ≠ b×a) | Rotations, spin |
| Octonions 𝕆 | 8 | Associativity ((a×b)×c ≠ a×(b×c)) | ??? |
This is Hurwitz’s theorem (1898): there is no five-dimensional, six-dimensional, or sixteen-dimensional version. Mathematics itself draws a hard boundary.
Each step in this sequence corresponds to a revolution in physics. Real numbers describe the classical world. Complex numbers describe quantum amplitudes. Quaternions describe spin and spatial rotations. The octonions have been waiting 180 years for their application.
64 Dimensions: Exactly One Particle Per Slot
The CHO framework takes the last three division algebras and combines them via tensor product:
𝒜 = ℂ ⊗ ℍ ⊗ 𝕆
This creates an algebra with 2 × 4 × 8 = 64 real dimensions.
This isn’t an abstract 64-dimensional space you’d need extra dimensions of spacetime to accommodate. It’s an internal space — the space of quantum numbers that label particles. Think of it as 64 slots, each corresponding to a specific type of particle state. Here’s how they map:
The 64 slots: one generation of the Standard Model
Each particle carries quantum numbers from each factor:
| Factor | Dimensions | What it encodes | Symmetry |
|---|---|---|---|
| ℂ (complex) | 2 | Particle vs antiparticle | U(1) — electromagnetism |
| ℍ (quaternion) | 4 | Weak isospin (left/right, up/down) | SU(2) — weak force |
| 𝕆 (octonion) | 8 | Colour (r, g, b, or colourless) | SU(3) — strong force |
Combining these gives 2 × 4 × 8 = 64 states per generation, which decomposes as:
Left-handed:
(ν_L, e_L) — lepton doublet 2 × 1 = 2 states (colourless)
(u_L, d_L) — quark doublet 2 × 3 = 6 states (3 colours)
Total left: 8
Right-handed:
ν_R — neutrino singlet 1 × 1 = 1 state
e_R — electron singlet 1 × 1 = 1 state
u_R — up quark singlet 1 × 3 = 3 states
d_R — down quark singlet 1 × 3 = 3 states
Total right: 8
Antiparticles: mirror of above 16 states
─────────
TOTAL: 32 complex = 64 real ✓
Every slot is filled. Every particle is accounted for. There is no room for extras.
This is why the framework predicts exactly the Standard Model particle content — no superpartners, no extra gauge bosons, no fourth generation. The algebra is saturated.
Three Generations: Count and Chirality, Without the Usual Obstruction
Every matter particle comes in three copies. The electron has heavier twins (muon, tau). The up quark has heavier twins (charm, top). The Standard Model accommodates this but doesn’t explain it.
In the CHO framework, three is structural — but the honest status is sharper than “it’s a theorem,” and getting it right took some care:
1. The tempting route — and why it’s blocked. The group Spin(8) that acts on octonions has a famous property called triality: an outer S₃ symmetry permuting three 8-dimensional representations. It is tempting to call those three the generations. But this is exactly the route closed by the Distler–Garibaldi no-go theorem: outer triality permutes three inequivalent modules (8_v, 8_s, 8_c) of mixed chirality, so it cannot deliver three same-chirality copies of one fermion. “Three from triality” does not work as stated.
2. The route that survives. Read the three generations instead as the three primitive idempotents of a Jordan frame in the exceptional Jordan algebra J₃(𝕆). The S₃ permuting them is inner — it lives inside the connected automorphism group F₄ — so it carries each idempotent to a congruent copy carrying the same self-conjugate spinor, and the obstruction above cannot even be posed. This makes the count (three) and the chirality (all alike) obstruction-free. What it does not yet deliver is the mass spectrum: the precise map from idempotents to fermion content remains an open bridge.
3. Why not a fourth. You can build a 3×3 Hermitian matrix algebra over the octonions (J₃(𝕆), dimension 27), but 4×4 provably fails — the identity that makes the Jordan structure work breaks down past 3×3 — and one further Cayley–Dickson step beyond the octonions produces the 16-dimensional “sedenions,” which contain zero divisors. The algebra has no room for a fourth like-chirality family.
So: the count and chirality of three generations rest on solid algebra (stated as standalone theorems, decoupled from any physics, in the project’s Jordan-theorems note); the spectrum — why these three carry the masses they do — is still an open derivation, not a settled result.
From Structure to Numbers
The algebra doesn’t just explain what exists — it calculates the properties of what exists.
A single parameter controls everything
The framework is organized around one key bridge target from the algebra:
\[\varepsilon_0^2 = \frac{\pi}{432} = \frac{\pi}{16 \times 27}\]where 16 = dim 𝕆P² (the real dimension of the Cayley projective plane F₄/Spin(9), equivalently the real Spin(9) spinor Δ₉) and 27 = dim(J₃(𝕆)) (the dimension of the exceptional Jordan algebra). Both factors are dimensions of specific geometric objects the framework derives, not numbers chosen by hand.
This gives ε₀ ≈ 0.0853. From this single algebraic bridge target — not treated as a fitted free parameter, but still awaiting its operator proof — the mass hierarchy and mixing pattern are audited.
The mass hierarchy
Each sector gets a different multiplicity factor from the algebra:
| Sector | 2nd/3rd gen ratio | Algebraic factor | Origin |
|---|---|---|---|
| Up quarks | m_c/m_t = ε₀² | ×1 | Single channel |
| Down quarks | m_s/m_b = 3ε₀² | ×3 | Three colours |
| Leptons | m_μ/m_τ = 8ε₀² | ×8 | Eight octonionic dimensions |
The factors 1, 3, 8 aren’t arbitrary — they’re the dimensions of the three “layers” of the algebra accessible to each particle type.
Relations vs experiment
With a few explicit inputs and bridge assumptions, and no row-by-row fitted low-energy parameters, the framework produces 23 grouped quantitative relations:
| What | Formula | Predicted | Measured | Error |
|---|---|---|---|---|
| Top quark mass | v/√2 | 174.1 GeV | 172.76 GeV | 0.8% |
| Higgs boson mass | v√(π/12) | 126.0 GeV | 125.09 GeV | 0.7% |
| Fine structure constant | 128π/3 + running | 1/137.0 | 1/137.036 | < 0.1% |
| Weinberg angle | 1/4 at Λ_QCD + RG | 0.231 | 0.23122 | < 0.1% |
| W boson mass | M_P/3³⁶ | 81.3 GeV | 80.4 GeV | 1.2% |
| Tau lepton mass | √2·ε₀²·m_t | 1.776 GeV | 1.777 GeV | 0.06% |
| Bottom quark mass | (7/3)·m_τ | 4.144 GeV | 4.18 GeV | 0.9% |
| Cabibbo angle (|V_us|) | √7·ε₀ | 0.2256 | 0.2243 | 0.6% |
| |V_cb| | ε₀/2 | 0.0426 | 0.0422 | 1.0% |
| PMNS θ₂₃ (forward bet — see below) | 4/7 | 0.5714 | octant unresolved | — |
| PMNS θ₁₃ | 3ε₀² | 0.0218 | 0.0220 | 1.0% |
| Neutrino Δm² ratio | 4ε₀² | 0.0291 | 0.0295 | 1.4% |
| CP violation (J_CKM) | NNI + arccos(1/3) | 3.01×10⁻⁵ | 3.08×10⁻⁵ | 2.3% |
| Cosmological constant | 3⁻²⁵⁶ suppression | ~2.3 meV | ~2.3 meV | ~3% |
| + 9 more | — | — | — | 0.2–7% |
Median descriptive error: 1.0%. All within 3σ experimental uncertainty for the 16 audit rows where experimental precision allows a meaningful pull calculation. These rows are correlated, so this is not a global independent-observable fit.
One of these is a forward prediction, not a postdiction. The PMNS angle θ₂₃ = 4/7 (upper octant) is not yet settled: the octant is experimentally unresolved (current T2K/NOvA results are in tension), so this row is a pre-registered bet that DUNE and Hyper-Kamiokande will decide — not a fit to a known value. It is the framework’s single sharpest falsifiable claim; the other rows are postdictions of already-measured quantities.
The Strong CP Problem: A Symmetry Argument, Not an Axion
One of the outstanding puzzles of particle physics: why does the strong force respect CP symmetry to such extraordinary precision? The parameter θ̄ that controls this is measured to be less than 10⁻¹⁰ — effectively zero. The Standard Model offers no explanation for this.
The leading proposed solution introduces a new particle (the axion) and a new symmetry. Decades of searches have found nothing.
In the CHO framework, θ̄ = 0 is a symmetry, not a coincidence:
- The Fano plane (the multiplication table of the octonions) has a Z₂ symmetry: reverse all seven directed lines simultaneously
- This Z₂ acts on the colour group SU(3) as charge conjugation: 3 ↔ 3̄
- Under this transformation, the θ-term flips sign: F∧F̃ → −F∧F̃
- Invariance of the algebra forces θ = 0
Meanwhile, the CKM phase that gives weak CP violation is preserved — it comes from a geometric angle between different Fano plane lines, which is unaffected by the Z₂.
This is a discrete-symmetry argument, and the project logs it as a derived bridge with an open obligation: showing that Fano parity is a genuine symmetry of the QCD path-integral measure (not merely the classical Lagrangian) has not yet been done. Treat it as a promising mechanism, not a closed proof.
Null target: no QCD axion. Axion searches should keep returning null results in the mass-coupling windows they cover; no single experiment covers the entire axion parameter space.
CP Violation: Why Matter Exists
The Big Bang produced matter and antimatter in almost-equal quantities. A tiny imbalance — roughly one extra matter particle per billion — is why the universe contains something rather than just radiation. This imbalance traces to “CP violation” in the weak force.
In the Standard Model, CP violation is parameterised by an angle δ that’s simply measured. The CHO framework derives it:
The Fano plane has seven lines, each containing three of its seven points. Two quaternionic subalgebras (one for up-type quarks, one for down-type) each sit on a line. Any two distinct lines of the Fano plane share exactly one point. The cosine of the angle between them:
δ = arccos(1/3) = 70.5°
This gives a Jarlskog invariant J = 3.01 × 10⁻⁵ (measured: 3.08 × 10⁻⁵). Your existence traces to the geometry of seven points and seven lines.
Dark Energy: The 10¹²² Problem
Quantum field theory predicts that empty space should contain enormously more energy than it actually does — off by a factor of roughly 10¹²². This “cosmological constant problem” has resisted solution for decades.
The CHO framework offers a candidate resolution: the vacuum energy is suppressed by 3⁻⁴ˣ⁶⁴ = 3⁻²⁵⁶ ≈ 10⁻¹²². Each of the 64 algebraic dimensions contributes a factor of 1/3 to the fourth power of the energy scale. The predicted dark energy density matches observation to ~3%.
Of all the framework’s numerical hits this is the least constrained: the exponent carries an extra O(1) prefactor that isn’t yet derived, and a look-elsewhere analysis shows that a simple “prefactor × power” form covers most of the available range — so this match carries less evidential weight than the dimensionless relations. Treat it as suggestive, not as a solution to the cosmological-constant problem.
Dark Matter: A Null Exclusion Target
The algebra is saturated — all 64 dimensions map to known particles. There is no algebraic “slot” for a new particle that carries Standard Model charges. Therefore:
- Dark matter is not a WIMP (no weak-force interaction)
- Dark matter is not a new particle in the usual sense
- It may be topological defects in the causal lattice structure — gravitationally interacting but otherwise invisible
This is consistent with 40 years of null results from direct detection experiments (LUX, XENON, PandaX, LZ). The useful future test is more specific: no confirmed WIMP-like recoil in the usual weak-scale mass window above next-generation experimental reach.
Neutrinos: Completing the Algebra
Right-handed neutrinos fill the last empty slot in the 64-dimensional algebra. Their Majorana mass scale:
M_R = M_Planck / 3⁹ ≈ 6 × 10¹⁴ GeV
Via the see-saw mechanism, this gives:
- m_ν₃ ≈ 49 meV (heaviest neutrino)
- Normal mass ordering (m₁ < m₂ < m₃)
- Sum of neutrino masses Σmᵢ ≈ 60 meV (frozen band 57–62 meV) — testable by the Euclid satellite
The large neutrino mixing angles (unlike the small CKM angles) are modeled by residual TBM symmetries in the Majorana sector, with broken-triality corrections from ε₀ bringing all three angles to sub-percent agreement with experiment. The exact residual symmetry still needs the operator-level derivation.
Spacetime from Information
The most speculative implication concerns spacetime. The framework posits a causal lattice — a discrete network of events, each carrying an algebraic label from ℂ⊗ℍ⊗𝕆. This is not yet a derivation of smooth spacetime, Einstein’s equations, or gravitational dynamics.
The non-associativity of the octonions plays a key structural role. In associative algebras, (a×b)×c = a×(b×c) always — there’s no “curvature” in how elements combine. Octonionic non-associativity means the order of combination matters. The current concrete result is a kinematic internal G2-covariant metric brick from the associator, not a 4D spacetime metric.
The Phase 5 audit keeps gravity outside the present framework: no canonical 4D Lorentzian reduction, field equation, or Newton constant has been derived.
What Would Falsify This Framework
Good theories expose themselves to clear failure modes. This framework is vulnerable to:
- Discovery of a 4th-generation particle (the algebra can’t support it)
- Proton decay observed (baryon number is algebraically exact)
- WIMP dark matter detected (no slot in the algebra)
- Inverted neutrino mass ordering (JUNO/DUNE, ~2028)
- Axion detected (the framework’s Fano-parity mechanism would be undercut)
- Higgs self-coupling far from λ = π/24 (HL-LHC, ~2030s)
Several of these directly contradict other popular theories: supersymmetry predicts superpartners; grand unification predicts proton decay; the axion hypothesis predicts an axion. These null claims are useful, but weaker than a positive quantitative prediction unless tied to explicit experimental reach.
What Would Strengthen It
- Normal neutrino mass ordering confirmed (JUNO, expected ~2028)
- Continued null results from WIMP and axion searches
- Neutrino mass sum Σmᵢ ≈ 60 meV (band 57–62 meV) (Euclid/DESI, ~2027–2030)
- Higgs self-coupling consistent with λ = π/24 (HL-LHC)
- Top quark mass measurements converging toward 174.1 GeV
Honest Caveats
This is a framework with intriguing initial results, not a finished theory:
- It is on probation, by its own scoreboard. Charged the full Occam price for every constant it chooses and credited only for those it derives, the project’s model-comparison Bayes factor currently sits at ln B ≈ −3 on closed results — i.e. mildly disfavoured — and only turns positive (≈ +6) if the geometric origin of ε₀² = π/432 is granted. The whole verdict hinges on one still-unbuilt derivation: an F₄-breaking action whose flux produces π/432. Until that exists, this is a hard-to-vary parametrization with strong structure, not an established theory.
- The continuum limit (showing smooth spacetime emerges from the discrete lattice) hasn’t been proven rigorously. This is a hard mathematical problem.
- The gravitational sector is explicitly out of scope for the present framework. There is a kinematic internal metric from octonionic non-associativity, but no 4D Lorentzian metric, Einstein equation, or Newton constant.
- The numerical relations are mostly tree level (lowest order). The 0.1–6% discrepancies should shrink when 1-loop corrections are computed from within the framework. This work hasn’t been done yet.
- The dark matter story is more “what it isn’t” than “what it is.” The algebraic-defect picture needs quantitative development.
- The m_e prediction (electron mass from 1st-gen formula) has the largest error at ~6%, suggesting the lepton NNI factor needs refinement or a proper loop calculation.
The Bigger Picture
For decades, attempts to go beyond the Standard Model have generally added structure — more symmetry (SUSY), more dimensions (string theory), more particles (dark sectors). These approaches introduce additional free parameters and have made few testable predictions.
The CHO framework goes in the opposite direction. It asks: what is the minimal mathematical structure from which the Standard Model must emerge? The answer turns out to be three division algebras combined in the only way they can be.
The result is a theory with fewer moving parts than the Standard Model (few explicit inputs rather than 19 fitted low-energy parameters), yet it reproduces — and in some cases extends — many of the Standard Model’s successful numerical relations. The 64 dimensions of ℂ⊗ℍ⊗𝕆 aren’t extra spatial dimensions to be hidden or compactified. They’re the internal quantum numbers of the particles we already know, organised by the deepest structure that mathematics allows.
Technical details live in the markdown foundation notes and executable audit scripts. The older LaTeX paper drafts have been removed; the authoritative form of the project is now code plus markdown.
This work was produced in collaboration with GitHub Copilot (ChatGPT 5.5 & Opus 4.8). The calculations, numerical checks, markdown notes, and derivation code are available on GitHub.