Neutrino NNLO Mass Ratios

τ-Formula

(Δpq, Δpr) = (8/7 + 3/175, 6/7 + 9/700) ≈ (1.16, 0.87)

Derivation

(Δ_pq,\, Δ_pr) \;=\; (87 + 3175,\; 67 + 9700) \;=\; (1.16,\; 0.87),

where the NNLO shifts are $δ_1 = 3/175 = 3/(7 · W_3(4)^2) = dim(τ^3)/(n · W_3(4)^2)$ and $δ_2 = 9/700 = 9/(n · 2|lobes| · W_3(4)^2)$, with $n = 7 = 2|lobes| + |sectors|$, $|lobes| = 2$, $|sectors| = 3$, $dim(τ^3) = 3$, $W_3(4) = 5$. Equivalently $700 = 7 · 100$ where $100 = 2|lobes| · W_3(4)^2 = 4 · 25$; every integer in the denominators is traceable to the $τ$-framework’s structural constants (no free integers). Three structural properties hold:

• 4/3 ratio preserved at every order: $Δ_pq/Δ_pr = 4/3$, where $4/3 = 2 · |lobes|/|sectors|$.

• Self-similarity: $δ_1/δ_0 = 3/25$, where $δ_0 = 1/7$ is the leading-order unit and $25 = W_3(4)^2$ is the window-squared normalization.

Source

This prediction is derived in the Numerical Physics Ledger (Chapter 61 — mixing-baryogenesis), Books IV–V of Panta Rhei.