Why 137? A Parameter-Free Derivation of Fundamental Constants from Modular Arithmetic
Joshua Christenson
March 2026 · v3
Core Discovery Paper
The foundational paper. Derives the fine-structure constant, proton-electron mass ratio, Weinberg angle, and 25 additional constants from three integers (n=9, g=2, N=8) and the golden ratio. All 28 constants match measurement to within 1.2σ. Zero free parameters.
1/α = Φ10 + 7Φ6/9 + 4/(9Φπ) + 1/(7Φ17) − 4/(5Φ34) → 137.035999 at 0.01σ
28 Constants
Zero Parameters
Sub-σ Precision
Z9 Casimir Structure: The Proton Mass Ratio as a Group Invariant
Paper II
2026
Proves that P = 1836 is a Casimir invariant of the Z9 grid structure. The eigenvalue spectrum λk = 9k² sums to 1836 across 8 depths. The second Casimir C⊂2; = 204 = Σk² connects the grid to hydrogen's shell degeneracy.
P = 9 × Σk² = 9 × 204 = 1,836 = mp/me
Mass Ratio
Casimir Invariant
Hydrogen Connection
Z9 Flavour Dynamics: A Lagrangian Realization
Paper III
2026
Constructs a Lagrangian for the three Z9 families {1,4,7}, {2,5,8}, {3,6,9}. Shows how the doubling circuit 1→2→4→8→7→5→1 generates flavour transitions and derives the CKM mixing structure.
Lagrangian
CKM Matrix
Three Families
Z9 Yukawa Coefficients: Rationalization and UV Completion
Paper IV
2026
Derives quark and lepton mass ratios as rational expressions in Z9 structural numbers. ms/md = 20, mb/ms = 45, md/mu = 17/8. All from {g, n, N, max147, max258}.
Quark Masses
Yukawa Couplings
UV Completion
Z9 and the Standard Model Gauge Group
Paper V
2026
Shows how the SU(3) × SU(2) × U(1) gauge structure emerges from Z9's three families and the doubling circuit. The 32 = g² × N fermion degrees of freedom per generation match exactly.
Gauge Group
SU(3)×SU(2)×U(1)
32 DOF
Z9 Phenomenology: Neutrino Predictions, Flavour Safety, and Experimental Tests
Paper VI
2026
Catalogues testable predictions: sin²θ23 = 4/7, normal mass ordering, m1 = 0. Identifies which experiments (DUNE, JUNO, KATRIN) can confirm or falsify each prediction.
DUNE
Neutrinos
Falsifiable
Z9 Paper VII: The Electron g-Factor from Grid Geometry
Paper VII
2026
Dedicated derivation of ae = 5/((162Φ³ − 5/(7Φ&sup7;) + 3/(8Φ¹&sup7;π)) × 2π). Shows each correction term uses Z9 family numbers with increasing Φ-power suppression. Matches CODATA at 0.6σ.
ae = 0.001 159 652 180 42 → 0.6σ from measurement
0.6σ
g-factor
3 Terms
Twenty-Eight Constants, Three Integers, and a Web That Cannot Be Faked
Information-Theoretic Proof
March 2026
Draft Complete
Proves that Z9 is not curve-fitting using information theory. Individual formulas use 602.6 bits to predict 349.8 bits — they don't compress alone. But the 21 structural relations produce 145 bits of free prediction at zero parametric cost. Combined p-value: 1.9 × 10⊃−4&sup0;. Thirty-three orders of magnitude beyond the Higgs discovery threshold.
p = 1.9 × 10−40 — 13.3σ — 145 bits of free compression
13.3σ
Kolmogorov Complexity
21 Relations
Numerology Killer
Z9 Gap Channel: Hydrogen Orbital Architecture from Modular Arithmetic
Gap Structure Paper
March 2026
Early Draft
The Yang-9 positions on the torus leave a continuous gap — a channel exactly n=9 columns wide at every depth, occupying fraction 1/k of the circumference. This reproduces hydrogen's orbital architecture from pure grid geometry. The 9s on each side of the gap form opposing scalar potentials that phase-conjugate across the channel — the Aharonov-Bohm region of the digital electron.
AB Region
Scalar Potentials
Hydrogen
Phase Conjugation
TorusNet: Z9-Native Neural Network Architecture for Infinite Context
AI Architecture
March 2026
Training
Tests whether Z9's toroidal memory structure can solve the context window problem. A fixed-size (16.9 MB) 8-depth torus state replaces the linear KV cache. Information propagates via the doubling circuit. Currently training on synthetic recall tasks targeting 50,000-token memory spans on a 512-token native window.
Neural Networks
Context Window
Toroidal Memory