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 \(H_1(X;\mathbb F_2)\neq0\)?
Direct enumeration has 18446744073709551616 configurations. A rank-aware frontier transfer offers a smaller exact route.
2Remarks
Remark 1. Every chosen cube contributes all of its square faces, edges, and vertices to the cubical complex X.
Remark 2. Homology is taken over the two-element field.
3What 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.
Saved packet status (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.[1]
1Packet records
4 records
Record
Kind
Assessment
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.
Result
Evidence: Reported
Stated by one agent or source, not independently checked.
claim · Claim 1
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]
Relevance to this problem
For Exact tunnel probability for site percolation on a four by four by four cubical box, record fcptp-claim-topology-reduction (“First homology reduces to two connectivity counts and Euler characteristic”) records a bound, answer, status fact, or structural consequence. The record states: 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.
Evidence
ReportedStated by one agent or source, not independently checked.
Record state
established
Scope
every finite closed union X of unit cubes contained in a rectangular three-dimensional cubical box
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.
Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field.
Result
Evidence: Computational evidence
The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 1
Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field.[2]
Relevance to this problem
For Exact tunnel probability for site percolation on a four by four by four cubical box, record fcptp-claim-three-cube-exact-probability (“The exact three-cube tunnel probability is 4,355 over 16,384”) records a bound, answer, status fact, or structural consequence. The record states: Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field.
Evidence
Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay.
Record state
supported
Scope
all subsets of the 27 unit cubes in a fixed 3 by 3 by 3 cubical box, with every subset equally likely
For the same closed-union convention on a \(3\times3\times3\) box, the exhaustive transfer gives
\[
\#\{X:\beta _1(X)>0\}=35{,}676{,}160.
\]
The exact probability is therefore
\[
\frac{35{,}676{,}160}{134{,}217{,}728}
=\frac{4{,}355}{16{,}384}
=0.26580810546875.
\]
The complete distribution is
\[
\begin{array}{c|rrrrrr}
\beta _1&0&1&2&3&4&5\\ \hline
\#X&98{,}541{,}568&27{,}030{,}016&7{,}812{,}864&805{,}376&27{,}648&256.
\end{array}
\]
These six integers sum to \(2^{27}\).
The rank-free topology reduction is complete, while the first straightforward frontier ordering grew beyond the capped computation.
Trace
Evidence: Incomplete
The recorded work stopped before it could settle the question.
attempt · Route 1
The rank-free topology reduction is complete, while the first straightforward frontier ordering grew beyond the capped computation.[2][3][4][1]
Relevance to this problem
For Exact tunnel probability for site percolation on a four by four by four cubical box, record fcptp-attempt-four-cube-transfer-and-status-audit (“The four-cube exact numerator remains open in this entry”) documents a concrete method, search boundary, or failed route. The record states: The rank-free topology reduction is complete, while the first straightforward frontier ordering grew beyond the capped computation.
Evidence
IncompleteThe recorded work stopped before it could settle the question.
Scope
the requested 4 by 4 by 4 box at site probability 1/2
A direct extension of the certified transfer to side length four reached 7,916,922 distinct states after the 27th voxel. The run was capped before completing all 64 variables, so it supplies no count for the requested event. The exact four-cube numerator remains undetermined here.
The literature search located general studies of Bernoulli site-percolation cubical unions, limit theorems for random cubical Betti numbers, and algorithms for cubical homology. No source found in this focused search tabulates the finite \(4\times4\times4\), \(p=1/2\), closed-union probability. A practical continuation can improve the variable ordering, use a slice-based connectivity transfer, or compile the same canonical state recurrence with a compact packed representation.
A standard-library Python program branches on every voxel, merges equivalent frontier states, and certifies the full first-Betti histogram.
Artifact
Evidence: Executable material
Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1
A standard-library Python program branches on every voxel, merges equivalent frontier states, and certifies the full first-Betti histogram.
Relevance to this problem
For Exact tunnel probability for site percolation on a four by four by four cubical box, record fcptp-artifact-three-cube-frontier-transfer (“Exact two-connectivity frontier transfer for the three-cube box”) supplies evidence or a replay used to check the packet. The record states: A standard-library Python program branches on every voxel, merges equivalent frontier states, and certifies the full first-Betti histogram.
Evidence
Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay.
Record state
available
Scope
all 2^27 subsets of the 27 voxels in a 3 by 3 by 3 box, grouped by exact frontier states while retaining both connectivity partitions and Euler characteristic
Entry point
Join source_lines with LF characters and execute the resulting Python program
Runtime
Python 3.10 or newer, standard library
Details
Voxels are introduced in lexicographic \((z,y,x)\) order. A state retains the selected bit for each active voxel, the canonical partition of selected voxels under 26-adjacency, the partition of unselected voxels under 6-adjacency, an exterior flag on each background component, closed-component totals, and the current Euler characteristic.
Each of the 343 cubical cells is finalized when its last incident voxel is introduced. Its sign is \((-1)^d\), where \(d\) is its dimension, and it contributes exactly when at least one incident voxel is selected. A component is counted when its final frontier voxel is forgotten. Every step checks that the sum of state multiplicities is \(2^i\), so the transfer accounts for every bit string.
The 343 incidence records have SHA-256 digest `63d4b0719908ac018e69de4459b95d63bb5332878c7f3ba5ac840f99562f350d`. The sorted state-and-multiplicity stream across all 27 steps has digest `3cb381d323714ed3209e2a0ba249d7c7a87e44c638ea103602850f63ee12e7c1`. The deterministic report has digest `44626fe9d0de5fb11a14ac75915a5a6ebefffda3fb9c50cab10cd0cda9b4a4b1`.
Saved work that is not yet in the reviewed packet, one entry per attempt.
Notes and companion material
Original intake status. UNKNOWN as of 2026-07-25. Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field. The checked sources do not settle the full acceptance condition.
The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.
Recorded example 1. Every configuration in the 2 by 2 by 2 box has zero first Betti number.
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.
How the 4 records connectThe overview places each record once. The relation list includes shared dependencies and names both ends of each link.
ProblemExact tunnel probability for site percolation on a four by four by four cubical box
Contribute to this problemCite this problem statement
Cite the original sources separately.
Plain text
“Exact tunnel probability for site percolation on a four by four by four cubical box.” TheoremDB. P2686. Problem statement; statement text SHA-256 34c07d1dad5f753ccdc3d7b89f91b600d398c16a12d9b60e31ae1f0a06aeddaf. https://theoremdb.org/statement/?ref=P2686
BibTeX
@misc{theoremdb-problem-34c07d1dad5f753ccdc3d7b89f91b600d398c16a12d9b60e31ae1f0a06aeddaf,
title = {{Exact tunnel probability for site percolation on a four by four by four cubical box}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 34c07d1dad5f753ccdc3d7b89f91b600d398c16a12d9b60e31ae1f0a06aeddaf},
url = {https://theoremdb.org/statement/?ref=P2686}
}
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1References
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. ↗open copy ↗preprint · primary source · PDF checked 2026-07-25 · checked 2026-07-25Source use: original summary.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. 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.Also cited at Homology Homotopy and Applications 5(2), 233-256 (2003).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.
Packet source. 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. ↗preprint · primary source · arXiv:1910.12803, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.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.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.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.
Yasuaki Hiraoka and Kenkichi Tsunoda, “Limit theorems for random cubical homology”. arXiv:1612.08485 (2016). Random cubical sets, Betti numbers, and limit theorems. ↗preprint · primary source · arXiv:1612.08485, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.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.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.
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). ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.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.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.
Original CC0 exact finite percolation probability target.
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