MASTER TEAPOTS AND ENTROPY ALGORITHMS FOR THE MANDELBROT SET

The paper proves Theorem 1.1—there is a continuous extension of Z+ from Q/Z to R/Z—by building the labeled-wedge graph Γθ for each angle, using its spectral determinant PΓθ, identifying roots in the unit disk with eigenvalues of appropriate finite models, and applying a Rouché-based continuity argument for zeros; this includes the crucial product/comparison with Hubbard-tree Markov matrices off the unit circle (Theorem 4.1) and an identification of roots in D between infinite and finite models (Theorem 5.6), culminating in Section 6’s continuity result and Lemma 6.4’s Hausdorff convergence statement . By contrast, the candidate solution asserts two steps that are not supported as stated: (i) for rational θ, it claims the spectral determinant of the infinite wedge equals det(I − tWθ) of the finite model, whereas the paper proves equality of roots in the disk, not equality of the functions (Theorem 5.6) ; and (ii) it asserts a strong shift equivalence RS and SR between the wedge matrix and the Hubbard-tree Markov matrix, while the paper instead proves a product relation PTh(t) = PMar(t)Q(t) with Q’s roots in {0} ∪ S1 (Theorem 4.1), which suffices to match eigenvalues off the unit circle but is weaker than strong shift equivalence . The continuity portion of the candidate solution closely mirrors the paper’s Rouché/compact-uniform-convergence approach, and its final conclusion is consistent with the paper, but the proof as written depends on claims the paper does not justify and that are not cited by the model.