# P2906: Computability of the area of the Mandelbrot set - ID: `P2906` - Reference: `mandelbrot-area-computable` - Page: https://theoremdb.org/statements/P2906 - Record maturity: Reviewed problem with recorded work ## Problem Let \(M=\{c\in\mathbb C:(z_m)_{m\ge 0}\text{ is bounded for }z_0=0,\ z_{m+1}=z_m^2+c\}\), and let \(A\) be its planar Lebesgue measure. Is \(A\) a computable real number? ### Definitions - **Definition.** A real number A is computable if an algorithm, given k>=1, returns a rational q with |A-q|<2^{-k}. - **Definition.** Planar Lebesgue measure is the usual two-dimensional area on C identified with R^2. ### What counts as a solution - Give an algorithm that, for every k>=1, returns a rational q with |A-q|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. - Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled. ## Status Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof. Exact unresolved remainder: Give an algorithm that, for every k>=1, returns a rational q with |A-q|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled. [2](#reference-2) [3](#reference-3) [1](#reference-1) [4](#reference-4) ## Work ### Evidence for the current status **Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof. Exact unresolved remainder: Give an algorithm that, for every k>=1, returns a rational q with |A-q|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled. The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof. Exact unresolved remainder: Give an algorithm that, for every k>=1, returns a rational q with |A-q|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled. ### Background and intake notes Certified coverings, escape-time bounds, and parameter-space decompositions can be reused at higher precision. The target isolates the computability of one geometric invariant rather than pixelwise membership in the set. - Original intake status: UNKNOWN as of 2026-07-27. The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof. - The MathOverflow page was checked on 2026-07-27, including its answer and all comments. The answer states that the area is computable under the conjecture that hyperbolic parameters have full measure and leaves the unconditional question open. - Hertling's work on computable representations of the Mandelbrot set and Dezotti's arXiv:1405.1933 were checked for a result about its area. They address computability and approximation phenomena without settling the exact real-number target stated here. - A later MathOverflow thread, question 453862, still asks whether the area is known and did not yield a proof of computability in the dated audit. - A local corpus search used Mandelbrot area, computable measure, and computable-analysis variants. It found no duplicate target. - Recorded example: Escape-time computation gives certified exclusion regions outside M, but a one-sided exclusion procedure alone does not compute A. ### Open directions - **Route 1** (reported): Give an algorithm that, for every k>=1, returns a rational q with |A-q|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled. [1](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `mandelbrot-area-computable`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. MathOverflow question 249617, “Computability of the area of the Mandelbrot set,” checked 2026-08-01. Question 249617, its answer, and every visible comment were checked on 2026-07-27. https://mathoverflow.net/questions/249617/is-the-area-of-the-mandelbrot-provably-computable - Also cited at Full question, answers, and visible comments concerning Computability of the area of the Mandelbrot set; checked 2026-08-01. - Also cited at Editorial research route recorded 2026-08-01. - forum; reference source; checked 2026-08-01 - Source use: citation_only - Source used to formulate or check the problem record. - Source used to assess the problem's recorded status. - For Computability of the area of the Mandelbrot set, the reviewed source scope is Full question, answers, and visible comments concerning Computability of the area of the Mandelbrot set; checked 2026-08-01.. The packet makes no inference beyond that cited scope. - Source named by the research packet. 2. Peter Hertling, “Is the Mandelbrot set computable?,” Mathematical Logic Quarterly 51(1) (2005), 5-18. DOI 10.1002/malq.200310124. main computable-analysis results and conditional criteria for the Mandelbrot set as a compact plane subset https://doi.org/10.1002/malq.200310124 - scholarly_publication; reference source; checked 2026-08-01 - Source use: citation_only - Source used to assess the problem's recorded status. - For Computability of the area of the Mandelbrot set, this source addresses computability of the set itself, a different output from computability of its Lebesgue area. 3. Alexandre Dezotti, “Convergence properties of the Gronwall area formula for quadratic Julia sets,” arXiv:1405.1933 (2014). abstract on failure of finite-sum approximation for Gronwall's filled-Julia-set area formula along the main-cardioid boundary https://arxiv.org/abs/1405.1933 - preprint; reference source; arXiv:1405.1933, checked 2026-08-01; checked 2026-08-01 - Source use: citation_only - Source used to assess the problem's recorded status. - For Computability of the area of the Mandelbrot set, this source was excluded as status evidence because it concerns individual filled Julia sets rather than computability of the Mandelbrot set's area; it is retained to document source-review history. 4. MathOverflow question 453862, “Computability of the area of the Mandelbrot set,” checked 2026-08-01. question statement and visible answers and comments distinguishing numerical estimates from proofs of the area https://mathoverflow.net/questions/453862/is-the-area-of-the-mandelbrot-set-known - forum; reference source; checked 2026-08-01 - Source use: citation_only - Source used to assess the problem's recorded status. - For Computability of the area of the Mandelbrot set, this source is a related live discussion and does not itself prove area computability.