Jacobsthal Eigenvalue Duality, the Rotating WdW Instanton, and Dark Matter

We present a derivation of the cosmological dark-matter density Ωdm = 2− √ 3 ≈ 0.268 from the Jacobsthal recurrence eigenvalues and the semiclassical treatment of the rotating Wheeler–DeWitt (WdW) instanton. Compared to v3, this version substantially strengthens the status of the central spin conjecture. Proved (unchanged): (i) The self-consistency equation x − 1/x = 3/2 has roots x+ = 2 and x− = −1/2; (ii) the rotating Bianchi IX instanton possesses a U(1) zero mode with collective-coordinate integral 2π. New in v4 — NCG derivation of the spin conjecture: The Seeley–DeWitt coefficient a2 for the biaxial Bianchi IX metric (Fan–Fathizadeh–Marcolli 2015) vanishes at the WdW nucleation point if and only if w3/w1 = 1/2. This condition is not imposed but follows from the WdW Hamiltonian constraint V = 0, which forces the static part of a2 to vanish exactly. The identification w3 = aK (NUT/Kerr parameter, Gibbons–Hawking 1979, Sec. 3) then gives aK/M = 1/2. The derivative corrections to a2 are shown to be irrelevant: the WdW constraint at the turning point eliminates all kinetic terms exactly. The conjecture is thereby reduced to a single citation-level identification and is elevated to a theorem conditional on the Taub-NUT/Kerr matching. Conditional theorem (unchanged): Ωdm = (J/SH) × 2π = 2 − √ 3, agreeing with Planck 2018 to 1.5%, zero free parameters, falsifiable by CMB-S4.