Undecidable Connectivity of Computable Fractal Limit Sets: Π⁰₁-Completeness, an Ω-Comb, and Incompleteness Thresholds

Can a fractal that a computer draws to any precision have an undecidable topology? The answer is yes, but undecidability appears only in the limit set and never at any finite resolution. We construct a uniformly computable family of planar compact sets. Each set consists of two self-similar frame carpets (dimension log 12/log 4) joined by a bridge, and the bridge is cut at scale 2^-s if and only if a Turing machine halts at step s. Together with an ε-chain criterion, this shows that connectivity of computable compact sets is exactly Π⁰₁-complete. The connectivity of every finite stage is decidable. A disconnection first becomes visible at a scale set by the halting time, which no computable function bounds. With S(5) = 47,176,870, this scale can lie far below the Planck scale even for five-state machines. A single computable "Ω-comb" encodes Chaitin's halting probability: tooth T_q remains attached to the spine if and only if Ω ≤ q. The component of the spine is a Peano continuum. Local connectedness fails only because infinitely many detached teeth accumulate at one point. Let F be a consistent theory that proves all true Σ₁ sentences. F proves every true detachment, but it proves no attachment below a threshold r_F > Ω, so infinitely many true attachments are unprovable. This zone of true but unprovable attachments has width at least 2^-(K(F) + O(log K(F))). If F is also Σ₁-sound, it decides all teeth at level n only if n ≤ K(F) + O(log K(F)). This corrects the intuition of a critical scale n* = O(|U|) beyond which local connectivity becomes undecidable. Decidability never fails at finite resolution. The only universal threshold concerns provability, and it is set by the complexity of the axioms. Physical cutoffs do not remove the undecidability of the idealised limit. The resource barrier for individual instances, such as six-state machines tied to Collatz-like problems, is a separate issue. Numerical illustrations use exact component labelling and binary lambda calculus approximations of a toy halting probability. They also include an enumeration of all 20,736 two-state Turing machines, which reproduces S(2) = 6. This is the sixth paper of a series. The earlier papers are 10.5281/zenodo.23189575, 10.5281/zenodo.23193512, 10.5281/zenodo.23194419, 10.5281/zenodo.23195881 and 10.5281/zenodo.23199122. Code, data and figures: https://github.com/Ruqing1963/fractal-connectivity-undecidability