Spectral Index n_s

τ-Formula

n_s = 1 − 2/N_e = 1 − 2/57 = 0.96491

Derivation

where $W_5(3) = 19$ is the Waring number—the minimum number of fifth powers of non-negative integers needed to represent any natural number as a sum of $3$-bounded summands. The number $19 = W_5(3)$ appears in the continued-fraction expansion of $ι_τ^-1$ as the $[5,3]$ window: the 5th-order CF coefficient evaluated at the 3rd structural depth. (Registry: V.D253, $τ$-effective, Wave 14A.)

Structural origin. The e-fold count is not a tuned parameter. It arises from two independent quantities:

• $(τ^3) = 3$: the dimension of the fibered product, itself decomposed as $1 + 2$ (base plus fiber).
• $W_5(3) = 19$: the CF window that governs the inflationary duration, encoding the number of primorial ticks at which the boundary characters remain above the exponential-expansion threshold.

The product $3 × 19 = 57$ falls within the range $50 ≤ N_e ≤ 60$ required by CMB observations to solve the flatness and horizon problems.

Source

This prediction is derived in the Physics Ledger (Chapter 62 — inflation-cmb-bbn), Books IV–V of Panta Rhei.