Finite-n Estimate of Dedekind Numbers Using Layer-Ratio Monte Carlo
Year: 2026
Type: article
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; the n=11 and n=13 center-window estimates show a higher-contrast analogous pattern. The protocol estimate for M(10) is M^(10)=(8.9360±0.0010)×1078, where the displayed uncertainty is the budget-based forecast scale from the cross-n scaling law under the production budget. It is an empirical variability scale associated with the stated Monte Carlo protocol and production budget, estimated from seed-level variation.
Source: Mathematics
DOI: 10.3390/math14183300
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7212303946