Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo

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.

Source: arXiv:2606.09795v1 - http://arxiv.org/abs/2606.09795v1 PDF: https://arxiv.org/pdf/2606.09795v1 Original Link: http://arxiv.org/abs/2606.09795v1