Periodic Table of Particles from Lattice Gravity

Mendeleev's periodic table organized 63 known elements into rows and columns and predicted three that had not yet been found. The organizing principle was simple: atomic number determines the row, valence determines the column, and empty cells are predictions. This paper does the same for the particles of matter.

We show that all known elementary fermions — electrons, neutrinos, quarks, and their heavier copies — can be organized into a table with the same logical structure. Rows are labeled by one quantum number (orbital angular momentum l), columns by another (generation number), and the electric charge of every particle follows from a single formula: Q = T₃ + Y/2 with Y = (−1)^{l+1}/(2l+1). Leptons sit in the first row (l = 0), quarks in the second (l = 1). The formula has one input — the integer l — and zero adjustable parameters. It reproduces every charge in the Standard Model exactly.

The table has a third row. At l = 2, the formula predicts 60 new particles that have never been observed: fermions with fractional charges +2/5 and −3/5, carrying five colors instead of three, confined into electrically neutral composites. These composites interact only gravitationally and fall in the mass range 2–100 GeV. They are dark matter candidates — and the table predicts them the same way Mendeleev's empty cells predicted gallium and germanium.

The classification arises from the Virtual Information Lattice (vIL) — a framework in which all of physics emerges from a single structural element: the edge weight w = exp(−2Φ) on a self-consistent three-dimensional cubic graph. The framework is defined by six axioms and yields 15 theorems connecting gravity, quantum mechanics, and particle physics. Each particle is a gravitationally self-trapped vibration (soliton) — a localized solution of the Schrödinger–Poisson system where matter generates a gravitational well and the well confines the matter. This self-trapping loop, studied for decades in the context of boson stars, gravitational atoms, and fuzzy dark matter halos, produces solutions naturally labeled by the standard quantum numbers (l, m_l, n_r, T₃). The lattice metric g_tt = −exp(−2Φ), g_rr = exp(+2Φ) is uniquely fixed by five elementary requirements: no horizons, locality, universality, the Newtonian limit, and flat spacetime in vacuum.

Beyond charges, the same framework produces several additional results without further input:

— Three generations from C(3,2) = 3 independent winding planes in three spatial dimensions, consistent with the LEP measurement N_ν = 2.984 ± 0.008.

— The Weinberg angle sin²θ_W = 1/4 at the Planck scale, running to 0.231 at the Z boson mass. The experimental value is 0.23122 ± 0.00004.

— Confinement from angular momentum closure: a state with Σm_l ≠ 0 carries a long-range multipole field whose energy diverges — it cannot be isolated. Only composites with Σm_l = 0 are free. This reduces the confinement problem — an open Clay Millennium Prize question — to an elementary multipole convergence theorem in three dimensions.

— The lepton mass ratio τ/e = 3468 from the variational action of the Schrödinger–Poisson system (experiment: 3477, error 0.3%), with the electron mass as the single calibration input. The Koide parameter Q_K = 2/3 is reproduced exactly. The muon mass is 6% off — the largest discrepancy, acknowledged as an open problem.

— The proton mass (938 MeV) as 99% gravitational binding energy when three l = 1 solitons overlap. Computed value: 961 MeV (error 2.4%).

The paper includes a systematic derivation of all 48 Standard Model states as proof by exhaustion (every particle, every quantum number, no exceptions), a visual poster of the complete table for quick reference, computed soliton profiles for the electron, muon, and tau, and a comparison with both the Standard Model and grand unified theories (SU(5), SO(10)). All numerical results can be verified with a calculator; a Python script with 66 automated checks is provided as supplementary material.

The framework makes three concrete falsification criteria. Discovery of a fourth neutrino generation would contradict C(3,2) = 3. Observation of proton decay at any rate would invalidate the topological stability argument. Measurement of fractional charges inconsistent with ±2/5 or ±3/5 in the 2–100 GeV range would rule out the predicted sector. If any of these is established experimentally, the framework in its present form is ruled out.

The vIL framework, its six axioms, 15 theorems with proofs, and the companion document with intuitive explanations are available from me under the same Zenodo record.