Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo
2026-06-08 • Information Theory
Information Theory
AI summaryⓘ
The authors look at a math problem called Dedekind's problem, which counts certain types of special functions or sets. They turn this counting into a problem about ratios of layers in a lattice and use Markov chains (a type of random process) to estimate these ratios. By testing their method on known cases, they improve confidence in their estimates and reveal new patterns in the data that were not obvious before. Their estimate for the next number in the series, M(10), is given with precise uncertainty based on their calculations.
Dedekind's problemmonotone Boolean functionsBoolean latticeWhitney numbersideal latticeMarkov chainsMonte Carlo methodsunimodal sequenceranked lattice
Authors
Tian-Shun Chen, Hao Feng, Haozhe Wang, Kilar Zhang
Abstract
Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number M(n). Backtests at M(8) and M(9) calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision n=9 estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; n=11 and n=13 center-window estimates show a larger-contrast analogous pattern. The protocol estimate for M(10) is \[ \widehat M(10)=(8.9360\pm0.0010)\times 10^{78}, \] where the displayed uncertainty is the budget-based forecast scale from the cross-n scaling law under the production budget.