α_s(M_Z) = Ω₄ / 23 = 0.117921
The real period of elliptic curve 1132b1, divided by the discriminant prime of x³ − x − 1, equals the strong coupling constant. Zero free parameters.
| Quantity | Value |
|---|---|
| Elliptic curve | E₄: y² = x³ + 4x + 1 (Cremona 1132b1) |
| Conductor | 1132 = 4 × 283 (Q-sector discriminant prime) |
| Real period Ω₄ | 2.71217511598... |
| Divisor | 23 = |disc(x³ − x − 1)| (ρ-sector discriminant prime) |
| α_s = Ω₄ / 23 | 0.117921 |
| PDG measured | 0.1179 ± 0.0009 |
| Error | 0.02% (0.023σ) |
No QCD. No renormalization group. No lattice calculation. A period divided by a prime.
pip install mpmath sympy
python3 pdt_strong_coupling.py
The Q-sector polynomial x⁴ − x − 1 defines the elliptic curve E₄ (Cremona 1132b1) whose real period is:
Dividing by 23 — the absolute discriminant of the ρ-sector polynomial x³ − x − 1 — gives the strong coupling constant at the Z mass:
The Birch and Swinnerton-Dyer conjecture is proved for this curve (Gross-Zagier and Kolyvagin, 1990). The period is an exact computable invariant. The match to experiment is within 0.023σ.
The two polynomials x³ − x − 1 and x⁴ − x − 1 sit at the Pisot boundary — the transition between convergent and divergent algebraic dynamics in the polynomial family xⁿ = x + 1. Their arithmetic properties are established in:
- Arithmetic Geometry at the Pisot Boundary — six theorems on discriminants, Galois groups, class fields, and the Hodge star (Zenodo)
- Elliptic Curves Associated to the Pisot-Boundary Polynomials — full arithmetic invariants of both curves (Zenodo)
The strong coupling constant is the first fundamental constant of nature identified with an exact computable invariant of arithmetic geometry, with zero free parameters.
The period can be confirmed against the LMFDB entry: https://www.lmfdb.org/EllipticCurve/Q/1132/b/1
Stephanie Alexander — stephanie@baryonix.com — 2026
Part of Pisot Dimensional Theory.