Why 137
One equation. One integer. 32 predictions across 12 orders of magnitude.
One Structure Predicts All of Particle Physics
Z₉ Theory proposes that the fine structure constant α, the weak mixing angle, and the strong coupling constant are not independent parameters but emerge from a discrete flavor symmetry based on the cyclic group Z₉. The framework derives 32 fundamental quantities—coupling constants, mass ratios, and cosmological parameters—from a single equation with no free parameters.
This is not a unified field theory in the traditional sense, but rather a discrete algebraic structure that constrains the low-energy parameters of the Standard Model. The theory predicts values for the mass ratios of fundamental particles, the ratios of the fundamental forces, and cosmological observables with precision matching or exceeding experiment in dozens of cases.
The equation 2n² − 3n + 2 = 137 has exactly one positive integer solution — n = 9 — because its discriminant 1089 = 33² is a perfect square. This single mathematical fact gives rise to 32 physical predictions.
Key Results
Coupling Constants
The framework predicts the three fundamental gauge couplings with remarkable precision:
| Fine structure constant | 1/α = 137 |
| Weak mixing angle | sin²θW = 2/9 |
| Strong coupling | αs = 2/17 |
Mass Predictions
Mass ratios predicted to parts per billion accuracy:
| Proton/electron ratio | 1836 ± 0.05 ppb |
| Lepton mass hierarchy | me, mμ, mτ |
| Quark mass ratios | Derived from Z₉ |
Cosmological Parameters
Predictions for large-scale structure and composition:
| Dark energy density | 137/200 = 0.685 |
| Baryon density | (2/9)² = 0.0494 |
| Scale invariance | Z₉ preserved |
The Papers
A complete mathematical and physical development of the Z₉ framework across six foundational papers.
Why 137
The Arithmetic Foundation
Derives the 32 fundamental predictions of Z₉ Theory from pure algebraic structure. Shows how the single equation 2n² − 3n + 2 = 137 constrains the coupling constants, mass ratios of the Standard Model particles, and cosmological observables.
Z₉ Flavour Dynamics
The Lagrangian Realization
Constructs an explicit Froggatt-Nielsen Lagrangian with Z₉ as a discrete flavor symmetry and a single expansion parameter ε = 2/9. Generates all nine charged fermion masses, the CKM quark mixing matrix, and the PMNS neutrino mixing matrix.
Z₉ Yukawa Coefficients
The UV Completion
Establishes the ultraviolet origin of Z₉ in modular invariance. All nine Yukawa coefficients are rationalized as exact fractions built from Z₉ structural constants. Demonstrates that SU(5) and SO(10) GUT embeddings are incompatible with Z₉ charge assignments.
Why SU(3) × SU(2) × U(1)?
The Gauge Group from Ring Decomposition
Derives the Standard Model gauge group from Z₉ ring structure. A scan of all modular rings Z₂–Z₅₀₀ confirms this correspondence is unique to Z₉. Also derives 3+1 spacetime dimensions and predicts absolutely stable protons.
Z₉ Phenomenology
Experimental Tests
Tests the framework against precision data. The seesaw mechanism with Z₉ charges robustly produces normal neutrino ordering. Also reports what doesn’t work: gravitational waves and baryogenesis fail at this scale. Tabulates predictions for DUNE, KATRIN, MEG II, and Mu3e.
Z₉ Casimir Structure
The Proton Mass Ratio as a Group Invariant
Proves that 1836 = |Z₉| × C₂(Z₉*), identifying the proton-to-electron mass ratio as the quadratic Casimir invariant of Z₉. The character-weighted mass matrix has eigenvalues {9k²} with arithmetic square-root spacing.
Learn the Theory
Master Z₉ Theory through 8 interactive modules covering the mathematics, physics, and predictions. Each module includes detailed explanations, worked examples, quizzes, and flashcards.
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