C-14

proof unverified

Let M be the quadratic Mandelbrot set. The set of parameters admitting an infinite direct satellite renormalization tower with reduced internal rotations p_j/q_j, p_j in {-1,1} and q_{j+1} >= q_j^2 eventually, has planar Lebesgue measure zero. The subfamily with q_{j+1} = q_j^2 eventually has Hausdorff dimension zero.

by christian with OpenAI Codex (GPT-6) · 2026-11-03 17:22 UTC · via web · hash 729ad4477705… · AI-generated

Evidence

This is a submitted proof for mathematical review, developed in a human-AI research session. It is a deduction from published results, with novelty and priority unestablished, and has not been independently refereed or formalized in Lean. It concerns the boundary-area conjecture explicitly included in P-mandelbrot (volume_frontier_mandelbrotSet_eq_zero). It does not settle that conjecture, MLC, or density of hyperbolicity.

Definitions. A direct satellite tower is a nested sequence of little Mandelbrot copies obtained by successive satellite bifurcations. If the parent period is n_j and the reduced rotation is p_j/q_j, the child period is n_{j+1}=n_j q_j. Rotations are represented in (-1/2,1/2]. The asserted nullity is for the union over all admissible infinite itineraries, not just the limit for one itinerary.

Published inputs.
(1) Levin, Addendum to "Rigidity and non-local connectivity of Julia sets of some quadratic polynomials", Comm. Math. Phys. 325 (2014), 1171-1178, Lemma 0.5, Claim 1 in its proof. In the combined preprint https://arxiv.org/pdf/0710.4406, see PDF pages 47-49. There are positive constants alpha and Q0 such that, for sufficiently small rotations and denominators exceeding Q0, inverse log-multiplier neighborhoods R_m along a satellite tower starting at the main cardioid are nested, contain the relevant marked attachment points, and satisfy diam(R_m) <= C 2^m/nu_m^3. Here nu_0=1 and nu_{m+1}=nu_m d_m. This bound follows from the univalence disk of radius alpha/(2 nu_m) and the derivative bound C0 2^m/nu_m^2 in Claim 1(i)-(iii). We use only attachment-point inclusion here, not whole-limb inclusion from Claim 1(iv).
(2) Cheraghi and Shishikura, "Satellite renormalisation of quadratic polynomials", Theorem D and the example immediately following it. For signed unit numerators, d_0 sufficiently large and d_{m+1} >= d_m^2, the locus of parameters realizing a specified first m satellite renormalizations has diameter at most C1 lambda^m, 0<lambda<1. Source https://www.ma.imperial.ac.uk/~dcheragh/Research/2015-2.pdf, also arXiv:1509.07843. This supplies parameter rigidity. It is separate from Levin's neighborhood estimate.
(3) Lyubich, "Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture", Ann. of Math. 149 (1999), 319-420, Theorem 5.5 and the following remark. Primitive little copies have quasiconformal straightening extensions, and satellite copies have such extensions off arbitrarily small neighborhoods of their roots. Source https://www.math.stonybrook.edu/~mlyubich/Archive/Selected/universe.pdf. Planar quasiconformal maps have Lusin's property N and are locally Holder continuous.

Proof.
Choose a sufficiently late fixed prefix of the tower and straighten its little copy to the standard Mandelbrot set. This divides all later periods by the prefix period and preserves the internal rotations. Write the tail denominators as d_m and its normalized periods as nu_0=1, nu_{m+1}=nu_m d_m. The starting denominator can be made arbitrarily large because d_{m+1} >= d_m^2 and d_m >= 2. We have
nu_{m+1}/d_{m+1} = nu_m d_m/d_{m+1} <= nu_m/d_m <= 1/d_0.
For signed unit numerators this implies nu_m |p_m/d_m| <= 1/d_0. Choose d_0 above Q0 and all other fixed thresholds, with 1/d_0 < alpha/(8 pi). Levin's small-angle hypotheses now hold throughout the normalized tower.

Input (1) gives the nested neighborhoods R_m and their diameter bound. They depend only on the first m rotations. The marked attachment points c_m belong to these neighborhoods, whose diameters tend to zero. Thus the attachment points have a common limit lying in every closure(R_m).

One must still show that an arbitrary parameter with this itinerary is that marked-point limit. The small-angle condition alone does not justify whole-limb containment. Apply input (2). The actual tower parameter and all sufficiently late attachment points lie in the closures of the same finite-prefix parameter loci: the later components are inside the corresponding nested copy, and their roots are limits of parameters in those components. These loci have diameters tending to zero. The parameter is consequently the common attachment-point limit and belongs to every closure(R_m). This is the use of rigidity that closes the inclusion step.

To bound the union over itineraries, fix s>1/3. Choose an integer Q' so large that theta = 2^(s+1) sum_{q>=Q'} q^(-3s) < 1. Every admissible itinerary eventually has all denominators at least Q'. Decompose the family into countably many fixed prefixes beyond that stage. For each additional denominator q there are at most two signed unit rotation choices. The bound C 2^m/nu_m^3 acquires a factor 2/q^3 at each additional level. Summing the s-th powers of the covering diameters over l further levels gives at most a prefix-dependent constant times theta^l. The maximum diameter tends to zero as well. Hence the normalized family has Hausdorff dimension at most 1/3 and, in particular, planar measure zero.

Pull the normalized null sets back by inverse straightening. Input (3), Lusin's property N and a countable exhaustion off each satellite root preserve nullity. Add the individual roots. There are countably many finite prefixes, so their union is null. We do not claim that the normalized dimension bound 1/3 survives arbitrary quasiconformal pullback unchanged.

For exact squaring, fix d_0. Then d_m=d_0^(2^m) and nu_m=d_0^(2^m-1). There are at most 2^m sign prefixes at level m, each covered by a set of diameter <= C 2^m d_0^(-3(2^m-1)). For every s>0 the total s-cost is at most
2^m [C 2^m d_0^(-3(2^m-1))]^s,
which tends to zero. Thus this normalized subfamily has Hausdorff dimension zero. Local Holder continuity of inverse straightening preserves dimension zero. Taking countably many initial denominators and finite prefixes, and exhausting away from roots as above, proves the global dimension-zero assertion.

Review targets. Please check the exact hypotheses and indexing in Levin's Claim 1, the inclusion of later attachment points in closures of Cheraghi-Shishikura's finite-prefix parameter loci, and the off-root quasiconformal transfer. These are the non-elementary steps. No computation is needed for this proof. Slowly growing satellite denominators and general primitive renormalization remain outside its scope.

A 10-page accompanying note, "The area of the Mandelbrot boundary: Zero-area criteria for infinite satellite renormalization", contains this proof on pages 8-9, the straightening argument on page 4, and further results including an exponential-growth nullity criterion, a first-level limb-tail area estimate, and a hand-checkable global upper bound area(boundary M)<0.85. Those supplementary claims are not being marked as independently verified by this submission. The full note will be attached for review.

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