Rigorous Spectral Parametrisation of Riemann Zeros via the Lambert W Function: An Unconditionally Valid Reconstruction Theorem

We establish an unconditionally valid, explicit parametrisation for theimaginary parts γn of the nontrivial zeros of the Riemann zeta function through theexact analytical inversion of the smooth Weyl counting function via the LambertW function.The base parametrisation Tbase(n) := 2π(n − 7/8)/W((n − 7/8)/e) inverts N¯exactly, satisfying N¯(Tbase(n)) = n − 7/8; the complete parametrisation F11(n) :=Tbase(n) + Tosc(n), where Tosc(n) := −S(Tbase(n))/ρ(Tbase(n)), satisfies the uncon-ditional absolute error bound|γn − F11(n)| ≤ 5.24(log log n)1/3for all n ≥ 3.The proof proceeds via a logically complete two-step bootstrap: the crude bound|γn − Tbase(n)| ≤ 20 is established without linearisation; the linearisation condi-tion is then verified; and the refined bound follows by cancellation of the log nfactor between ϕ(γn) and ρ(ξ). The analytically-derived limiting constant is2π × 0.224 = 1.408. Numerical validation against 106 high-precision zeros confirmsall theoretical predictions. All results are unconditional: no assumption on theRiemann Hypothesis is made.