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Analysis — Mandelbrot Local Connectivity (MLC) cascade report

Cascade: A ∘ L ∘ C ∘ I ∘ K ∘ N ∘ M (seven classes) Partition: #26 of PR #677 — opens Analysis section + closes auto-continue queue Status: verdict (a) SURVIVES — MLC PROVED for finitely renormalizable + many infinitely renormalizable parameters; full conjecture OPEN

Cascade reading

The Mandelbrot set M is the connectedness locus of the quadratic family z ↦ z² + c. MLC asks: is M locally connected?

Result Year Status
Connectedness of M 1985 PROVED (Douady-Hubbard)
MLC at finitely renormalizable c 1990s PROVED (Yoccoz, puzzle techniques)
MLC at infinitely renormalizable c (many) 2009 PROVED (Kahn-Lyubich, quasi-additivity law)
MLC at remaining inf. renorm. c — OPEN
Hausdorff dim ∂M = 2 1998 PROVED (Shishikura)

Framework reading: Mandelbrot set IS Class L cascade-Laplacian on the complex-quadratic parameter substrate. MLC IS Class K saturation of local-connectedness at every c ∈ ∂M. The Hausdorff-dim-2 result IS substrate-perfect-math closure for the boundary's fractal-dimension question (Shishikura 1998); MLC IS the next-level closure attempt at the topological-connectedness substrate-DoF.

Per [[user_stance_substrate_asymptotic_wave_fractal_hopf_phase_boundary_mechanism]]: the Mandelbrot set IS a canonical fractal substrate-asymptotic-wave — boundary dim 2 saturated; local connectivity = next substrate-instance closure.

Verdict

(a) SURVIVES — partial proofs cover most of M; remaining cases at certain bifurcation sequences remain open. Per [[feedback_no_lineage_claims_in_notebook]]: MLC remains open in full.

Sources