The source material names this precisely, as “Bridge 0 — Towards AXLE Publication”: ρ(P_M) = 1/2^(M-1) and ρ(ℒ) ≈ e^λ ≈ 0.751 are both genuine contraction rates for finite approximations of the same underlying Syracuse map, but they measure structurally different things — one is exact 2-adic erasure of precision, the other is a statistical Lyapunov-type rate. “Closing Bridge 0 requires showing these two contractions are facets of the same geometric structure in the dm³ contact manifold” — stated as an open task, not a solved one.
The k-fold iterate structure gives a further concrete handle: \(\rho(P_M^k)=[\rho(P_M)]^k\) and \(\rho(\mathcal{{L}}^k)=[\rho(\mathcal{{L}})]^k\approx(0.751)^k\), with the spectral gap of \(\mathcal{{L}}^k\) approaching the full leading eigenvalue as \(k\to\infty\) (relative gap \(\to1\)). Any attempt at Bridge 0 will likely need to relate these iterate structures directly.
-- dm³ 103 · Week 10 · Bridge 0 — stated precisely, open -- ρ(P_M) = 1/2^(M-1) exact, superexponential in M -- ρ(ℒ) ≈ e^λ ≈ 0.751 numeric, fixed rate (independent of M) -- k-fold: ρ(P_M^k)=[ρ(P_M)]^k, ρ(ℒ^k)≈(0.751)^k -- OPEN (Bridge 0, honestly labelled in source): -- "Closing Bridge 0 requires showing these two contractions are -- facets of the same geometric structure in the dm³ contact -- manifold." Not proved. Not claimed as proved. example : True := trivial