Navier–Stokes Regularity (No Blow-Up)
Navier–Stokes Regularity (No Blow-Up): τ-value no blow-up, observed (open), deviation –.
Prediction
τ-Formula
Profinite decompactification → no finite-time blow-up
Derivation
where $p_n^#$ is the $n$th primorial. The convergence to the Leray exponent $α = 1$ is super-exponential:
| $n$ | $p_n^#$ | |||||||||||||||||
| $α_n$ | ||||||||||||||||||
| Gap $1 - α_n$ | ||||||||||||||||||
| 1 | 2 | 0.500 | $5.0 × 10^-1$ 2 | 6 | 0.833 | $1.7 × 10^-1$ 3 | 30 | 0.967 | $3.3 × 10^-2$ 4 | 210 | 0.995 | $4.8 × 10^-3$ 5 | 2310 | 0.9996 | $4.3 × 10^-4$ 6 | 30030 | 0.99997 | $3.3 × 10^-5$ |
By depth $5$, the exponent is within $0.04%$ of the Leray value. The gap closes super-exponentially: the primorials grow faster than any exponential.
The gap to the Millennium Problem. The decompactification theorem shows that the $τ$-regularity exponent converges to the Leray value faster than any geometric sequence. This is strong structural evidence for Navier–Stokes regularity, but it is not a proof in the Clay Mathematics Institute sense: the Millennium Problem requires regularity on $ℝ^3$, not on a sequence of compact approximations. The scope remains conjectural (Chapter (ch:book5-ch27-navier-stokes-macro), Section (sec:ch27-clay-bridge)).
Source
This prediction is derived in the Physics Ledger (Chapter 65 — collective-dynamics), Books IV–V of Panta Rhei.