Interactive
Two integers in, numbers out
These are the framework's actual closed-form expressions, evaluated in your browser. Change p or z and see how sensitive each result is. The defaults are the values the framework uses.
z = 6
z = 6 is the cubic tiling of the two-sphere. Other values are not a BPR configuration; they are shown only to display sensitivity.
Inverse fine-structure constant(ln p)² + z/2 + γ − 1/2π · derived, p anchored
137.035999
Solar mixing sin²θ₁₂1/3 − 1/(3.5 ln p) · framework
0.3092 ± 0.0087
Reactor angle θ₁₃sin θ₁₃ = e⁻¹/√(2·3) · derived, ngen input
8.54° ± 0.15°
CKM CP phase δπ/2 − 1/√(z+1) · derived
68.5° ± 5.7°
mμ/mez(z²−1) · conjectural
206.77
mτ/me(z(z+4)−1)² · conjectural
3477.2
ms/mdfrom labels (1, z−2) with shift Wc = √3 · conjectural
20.0 ± 1
Top-quark label ℓt(z²−1)(z+5−z/2)+3 · conjectural, index argument withdrawn
—
Lorentz violation |δc/c|quadratic parameter 1/p, scaled · derived
< 6 × 10⁻²¹
ΔNeff ceilingp1/3 · derived
CMB-S4 2030+
The lepton and quark rows use the flavor ansatz whose derivation was withdrawn in September 2026. The formulas still produce the numbers; that is all they do. The Lorentz-violation row uses the quoted scaling from the roadmap and is approximate.
What to notice
- 1/α depends on p only through ln p, so it is insensitive: a factor of 2 in p moves it by about 16. That is why four unrelated relations can all "point to" a prime near 10⁵ without any of them being precise.
- The angles that depend only on z (the CP phase) do not move when you change p at all. The framework's most striking agreements are its least adjustable ones.
- The mass ratios are pure combinatorics of z. They are the easiest thing on this page to get suspicious about, and the project now agrees.