# P2686: Exact tunnel probability for site percolation on a four by four by four cubical box - ID: `P2686` - Reference: `four-cube-site-percolation-tunnel-probability` - Page: https://theoremdb.org/statements/P2686 - Record maturity: Reviewed problem with recorded work ## Problem Choose each of the \(64\) unit cubes in a \(4\times4\times4\) box independently with probability \(1/2\), and let \(X\) be their closed union. What is the exact probability that \(H_1(X;\mathbb F_2)\neq0\)? ### Remarks - **Remark.** Every chosen cube contributes all of its square faces, edges, and vertices to the cubical complex X. - **Remark.** Homology is taken over the two-element field. ### What counts as a solution - Give the exact numerator and denominator, with a complete transfer, decision-diagram, or exhaustive certificate whose boundary matrices can be replayed. ## Status The identity \(\beta_1=c_{26}+h_6-\chi\) and the exact \(3\times3\times3\) probability \(4355/16384\) are established, while the capped \(4\times4\times4\) transfer stopped after 27 of 64 voxels; the requested four-cube numerator remains undetermined. [1](#reference-1) ## Work ### Evidence for the current status **Theorem 1 (First homology reduces to two connectivity counts and Euler characteristic).** The identity \(\beta_1=c_{26}+h_6-\chi\) and the exact \(3\times3\times3\) probability \(4355/16384\) are established, while the capped \(4\times4\times4\) transfer stopped after 27 of 64 voxels; the requested four-cube numerator remains undetermined. Let \(c_{26}(X)\) be the number of components of the chosen voxels under vertex-touching adjacency. Let \(h_6(X)\) be the number of face-connected components of unchosen voxels that miss the boundary of the box. Then \[ \beta _1(X;\mathbb F_2)=c_{26}(X)+h_6(X)-\chi(X). \] The closed cubes that meet at a face, edge, or vertex belong to the same connected component, which gives \(\beta _0(X)=c_{26}(X)\). The bounded components of \(\mathbb R^3\setminus X\) correspond to face-connected unchosen-voxel components that do not meet the box boundary. Alexander duality gives \(\beta _2(X)=h_6(X)\), and \(\beta _3(X)=0\) for a bounded subset of \(\mathbb R^3\). Substitution in \[ \chi(X)=\beta _0(X)-\beta _1(X)+\beta _2(X) \] proves the formula. The transfer computes \(\chi\) directly as the number of present vertices minus present edges plus present faces minus selected cubes. ### Background and intake notes Direct enumeration has 18446744073709551616 configurations. A rank-aware frontier transfer offers a smaller exact route. - Original intake status: Novelty remains unverified. No primary-source status audit was completed for this exact box and boundary convention. - A transfer state should retain the frontier cubical incidence data needed for rank updates, rather than a floating Euler characteristic. - Euler characteristic alone does not determine the first Betti number. Digital foreground and background connectivity conventions can also give wrong tunnel counts. - The exact answer has denominator 2^64 before reduction. Complement and box symmetries can pair configurations. - Recorded example: Every configuration in the 2 by 2 by 2 box has zero first Betti number. ### Other known results - **Computation 1** (reproduced): Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field. [3](#reference-3) ### Prior approaches - **Route 1** (reproduced): The rank-free topology reduction is complete, while the first straightforward frontier ordering grew beyond the capped computation. [3](#reference-3) [2](#reference-2) [4](#reference-4) [1](#reference-1) ### Runnable artifacts - **Artifact 1** (reproduced): A standard-library Python program branches on every voxel, merges equivalent frontier states, and certifies the full first-Betti histogram. ### Computational notes - Boundary matrices were built as bitsets and reduced exactly over F_2. Exhaustive enumeration of all 256 configurations of the 2 by 2 by 2 box found beta_1=0 throughout. For the target, 50000 configurations from seed 20260724 gave 48569 with beta_1>0, an empirical frequency of 0.97138; observed beta_1 values ranged from 0 through 19. ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `four-cube-site-percolation-tunnel-probability`, 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. Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek, “Computing Homology,” Homology, Homotopy and Applications 5(2) (2003), 233-256. Kaczynski, Mischaikow, and Mrozek, Computing Homology, Homology Homotopy and Applications 5(2), 233-256 (2003), for cubical chain complexes; the displayed reduction also uses Euler-Poincare and Alexander duality https://archive.intlpress.com/site/pub/files/_fulltext/journals/hha/2003/0005/0002/HHA-2003-0005-0002-a008.pdf - Also cited at Homology Homotopy and Applications 5(2), 233-256 (2003) - website; reference source; PDF checked 2026-07-25; checked 2026-07-25 - Open copy: https://math.uchicago.edu/~shmuel/AAT-readings/Data%20Analysis%20/Mischaikow%2C%20computing%20homology....pdf - Source use: citation_only - For Exact tunnel probability for site percolation on a four by four by four cubical box: The identity \(\beta_1=c_{26}+h_6-\chi\) and the exact \(3\times3\times3\) probability \(4355/16384\) are established, while the capped \(4\times4\times4\) transfer stopped after 27 of 64 voxels; the requested four-cube numerator remains undetermined. 2. Yasuaki Hiraoka and Kenkichi Tsunoda, “Limit theorems for random cubical homology”. arXiv:1612.08485 (2016). Random cubical sets, Betti numbers, and limit theorems https://arxiv.org/abs/1612.08485 - preprint; reference source; arXiv:1612.08485, version checked 2026-07-25; checked 2026-07-25 - Source use: citation_only - For Exact tunnel probability for site percolation on a four by four by four cubical box: The four-cube exact numerator remains open in this entry. The rank-free topology reduction is complete, while the first straightforward frontier ordering grew beyond the capped computation. 3. Kenneth Dowling and Erik Lundberg, “Homotopy Types of Random Cubical Complexes”. arXiv:1910.12803 (2019). Kenneth Dowling and Erik Lundberg, Homotopy Types of Random Cubical Complexes, for the site-percolation cubical-union model; Bernoulli site percolation on a cubical grid and homotopy-type limit laws https://arxiv.org/abs/1910.12803 - Also cited at Bernoulli site percolation on a cubical grid and homotopy-type limit laws - Also cited at Exact symbolic enumeration in fcptp-artifact-three-cube-frontier-transfer - Also cited at Kenneth Dowling and Erik Lundberg, Homotopy Types of Random Cubical Complexes, for the site-percolation cubical-union model - preprint; reference source; arXiv:1910.12803, version checked 2026-07-25; checked 2026-07-25 - Source use: citation_only - For Exact tunnel probability for site percolation on a four by four by four cubical box: The rank-free topology reduction is complete, while the first straightforward frontier ordering grew beyond the capped computation. - Source named by the research packet. 4. Tomasz Kaczynski and Marian Mrozek, “The Cubical Cohomology Ring: An Algorithmic Approach”. Foundations of Computational Mathematics 13(5) (2013), 789-818. DOI 10.1007/s10208-012-9138-4. Foundations of Computational Mathematics 13, 789-818 (2013) https://doi.org/10.1007/s10208-012-9138-4 - scholarly_publication; reference source; version of record; checked 2026-08-01 - Source use: citation_only - For Exact tunnel probability for site percolation on a four by four by four cubical box: The four-cube exact numerator remains open in this entry. The rank-free topology reduction is complete, while the first straightforward frontier ordering grew beyond the capped computation.