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REVIEW 3 major objections 5 minor 14 references

Statistical Monte Carlo methods give D(10)–D(15) to 4–2 significant digits, far tighter than prior asymptotics.

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2026-07-10 07:20 UTC pith:H454E5IW

load-bearing objection Solid multi-method Monte Carlo estimates for D(10)–D(15) that beat Korshunov and match concurrent D(10) work; the soft spot is that for n≥13 the midpoint discrepancy exceeds the quoted SE, so the 2-digit claims rest on an unquantified systematic. the 3 major comments →

arxiv 2607.08446 v1 pith:H454E5IW submitted 2026-07-09 math.CO

Statistical Estimation of higher Dedekind Numbers

Alex Fihman , Lennart Van Hirtum , Christian Plessl This is my paper
classification math.CO MSC 06A0705A1568R0568W2065C05
keywords Dedekind numbersmonotone Boolean functionsantichainsMonte Carlo methodsMCMCcombinatorial enumerationweight layer branching
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Dedekind numbers D(n) count the monotone Boolean functions on n variables; exact values stop at D(9) because the sequence grows double-exponentially. This paper supplies the first high-accuracy statistical estimates for D(10) through D(15), reporting four reliable digits for D(10) and two for D(15). Three independent sampling techniques—pair matching of random 9-variable functions, reference-subset hit rates, and weight-layer branching—are used; where more than one method applies they agree inside their standard errors. The results replace Korshunov’s asymptotic formulas, which err by tens of percent even on known values, and give concrete numerical targets for future exact or improved Monte Carlo work. A sympathetic reader cares because Dedekind’s problem is a classical open enumeration challenge whose next few terms have been out of reach for decades.

Core claim

Weight-layer branching, cross-checked by pair matching and reference-subset sampling wherever feasible, yields D(10)≈8.93345×10^78 (S.E. 2.44×10^74) through D(15)≈3.80603×10^1953 (S.E. 5.30×10^1951). Multi-method consistency for n=10–12 and midpoint discrepancies of order 10^{-4}–10^{-2} for higher n support the claim that these figures are accurate to the stated precision and substantially better than earlier estimates.

What carries the argument

Weight Layer Branching: degree-corrected random walks confined to adjacent weight layers of the MBF graph measure average left/right branching ratios; the ratios are propagated from the constant-0 and constant-1 functions to the middle layer, producing estimates of every layer cardinality and therefore of D(n).

Load-bearing premise

For n≥13 the estimates rest on the premise that short degree-corrected walks between neighbouring weight layers have mixed enough for the observed branching ratios to represent the true global layer sizes.

What would settle it

An independent high-precision computation of D(10)—exact enumeration or a wholly different Monte Carlo scheme—that lands outside the interval 8.93345×10^78 ± a few reported standard errors would falsify the central numerical claims.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Four-digit D(10) and two-digit D(15) replace Korshunov formulas whose relative errors exceeded 50 percent on known values.
  • Agreement among three methods for D(10)–D(12) cross-validates the MCMC mixing diagnostics and degree correction.
  • Midpoint discrepancies of order 10^{-4}–10^{-2} supply a concrete residual-error diagnostic for the higher estimates.
  • The same sampling infrastructure immediately supports the larger reference subsets and Metropolis–Hastings variants listed as future work.
  • Layer probabilities obtained en route yield an independent formula for expected Hamming distance between random MBFs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Middle-layer concentration of min-positive and max-negative sets for n>9 suggests asymptotic formulas can treat only those layers as free variables.
  • Degree-corrected MCMC on the MBF graph is likely transferable to free distributive lattices and other graded posets of similar growth.
  • If midpoint discrepancy grows only slowly, the method can reach D(16)–D(18) with feasible core-years before exact enumeration becomes realistic.
  • The sequence of pair-matching probabilities p_match,n itself may admit an independent asymptotic analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript reports Monte Carlo estimates of the Dedekind numbers D(10)–D(15), improving substantially on Korshunov’s asymptotics and on concurrent layer-ratio work for D(10). Three estimators are used: (i) pair-matching frequency of uniform MBF9 samples to obtain D(10); (ii) MCMC hit rates into closed-form 1-layer reference subsets for D(10)–D(12); and (iii) degree-corrected weight-layer branching, which supplies estimates for all of D(10)–D(15). For n=9 the MCMC is validated against large uniform pair-matched samples (KS p=0.29 after 30k burn-in); Hamming-distance and weight-balance diagnostics are given for n=9–10. Multi-method agreement within reported S.E. is shown for D(10)–D(12) (Tables 11–13). For D(13)–D(15) only weight-layer branching is practical; Table 14 lists values, run-to-run S.E., core-day runtimes, and midpoint discrepancies between upward and downward layer propagations.

Significance. Exact Dedekind numbers are known only through D(9); higher values are of lasting interest in enumerative combinatorics and order theory. The paper supplies the first estimates with explicit numerical uncertainties for D(10)–D(15), multi-method cross-checks where feasible, open implementations, and a clear improvement over the only previously available asymptotic formulae (errors of tens of percent). The concurrent independent D(10) estimate of Chen et al. is consistent with the present results, which strengthens confidence in the n=10 figure. If the higher-n error budgets are made rigorous, the tables will become standard reference values for the community.

major comments (3)
  1. [Section 4.4, Table 14] Section 4.4 and Table 14: the reported S.E. for D(13)–D(15) is taken solely from run-to-run variation of the branching ratios (relative error on the most populated layer). For n=15 the midpoint discrepancy between the independent upward and downward propagations is 2.90×10^{-2}, roughly twenty times larger than the quoted relative S.E. (~1.4×10^{-3}). The discrepancy is therefore a consistency diagnostic that exceeds the published uncertainty, indicating residual systematic bias (incomplete local mixing on middle layers, or imperfect degree correction inside the two-layer subgraphs) that is not folded into the error budget. The abstract’s claim of “2 digits for D(15)” and the S.E. column of Table 1 rest on this unquantified component. Either enlarge the uncertainty to cover the observed discrepancy (or a calibrated multiple of it), or supply independent diagnostics that demonstrate the d
  2. [Sections 3.4.2–3.5, 4.4] Sections 3.4.2–3.5 validate mixing (Hamming distance, weight-balance, KS against uniform samples) only for n=9 and, partially, n=10. For n≥13 the estimator relies entirely on degree-corrected walks on adjacent weight layers (Section 4.4). The paper should state what burn-in, thinning, and local-mixing checks were used at those dimensions, or acknowledge that the n=9–10 diagnostics are being extrapolated. Without this, the assumption that the left/right branching ratios faithfully represent true layer cardinalities remains the weakest link for the D(13)–D(15) claims.
  3. [Table 1, Tables 11–13] Table 1 and the abstract present a single “best” value and S.E. for each n. For D(10)–D(12) the three methods differ by amounts comparable to (or larger than) the smallest quoted S.E. (e.g., reference-subset D(12)=7.1911×10^{283} vs weight-layer 7.1492×10^{283}). The final reported figure should either be a variance-weighted combination with an enlarged uncertainty that reflects method-to-method scatter, or the text should explicitly justify why one method’s S.E. is preferred over the inter-method spread.
minor comments (5)
  1. [Table 3] Table 3 caption states that relative errors for D(10)–D(15) are “relative to this paper’s estimates … and [are] therefore [themselves] uncertain.” That is correct, but the table still prints those percentages in the same column as exact-error percentages for D(0)–D(9). A visual distinction (e.g., italics or a second column) would avoid over-reading the higher-n entries.
  2. [Table 9, Theorem 3] Equation (13)–(14) and Table 9: p_match,n for n≥9 are derived backwards from the paper’s own D(n) estimates. The table header “Best currently known values” is slightly misleading for those rows; label them as “inferred from Table 1” to avoid circular appearance.
  3. [Figure 2] Figure 2 caption: “The relative height between series has no meaning” is clear, but the arbitrary vertical scaling constants are not stated; a short note that each curve is independently normalised would help reproducibility of the figure.
  4. [Table A.15] Appendix A.15 reports average antichain sizes up to n=15 with S.E.; the n=15 entry has S.E. 0.767 on a mean of ~2295, which is fine, but the text never uses these numbers outside the filter-tree discussion. Either cite them in the main MCMC section or move the higher-n rows to a repository note.
  5. [Throughout] Minor typos: “soft page faults” (p. 5) is fine; “unneeded MBF6 samplings” → “unnecessary”; “the bulk of the time” is colloquial for a journal. “S.E.2.44 imes10^{74}” in Table 1 needs a space after “S.E.”.

Circularity Check

0 steps flagged

No circularity: Monte Carlo ratio estimators (pair-match probability, closed-form reference hit rate, layer branching factors) are independent of the target D(n) values they produce.

full rationale

The paper estimates D(10)–D(15) by three sampling procedures whose outputs are combinatorial frequencies, not algebraic rearrangements of fitted constants. Pair matching (Thm. 3, Eq. 13–14) measures the empirical success rate of combining uniform MBF9 samples and multiplies by the independently known exact D(9)^2; the known D(n) for n≤8 and the R(7) equivalence-class table are used only as generation primitives or validation baselines, never as the quantity being estimated. Reference-subset sampling (Sec. 4.3, Eq. 16) counts MCMC hits into the closed-form set of 1-layer middle-layer functions whose cardinality is exactly 2^{binom(n,k)}; D(n) is recovered as |S|/p̂_S after standard degree correction π∝deg. Weight-layer branching (Sec. 4.4) estimates left/right degrees on adjacent weight layers by degree-corrected random walks, propagates the ratios from the two constant functions to the middle layer, and sums the resulting layer cardinalities; the midpoint discrepancy is reported as a consistency diagnostic, not as an input that forces the estimate. MCMC mixing is validated against independent uniform pair-matched samples for n=9 (KS p=0.29) and against long-run statistics for n=10; the concurrent Chen et al. layer-ratio result is an external cross-check. No step reduces a claimed prediction to a fitted parameter or to a self-citation that itself encodes the target. The derivation chain is therefore self-contained Monte Carlo measurement.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The work is computational Monte Carlo enumeration on a well-defined combinatorial object. Load-bearing background is standard: definition of MBFs/antichains, pair-matching bijection (Lemma 1 / Corollary 2), undirected MBF graph with π∝deg, bipartiteness by weight parity, and closed-form size of 1-layer middle-layer subsets 2^{C(n,⌊n/2⌋)}. Algorithmic knobs (burn-in length, block size, split threshold) are engineering choices, not fitted to the Dedekind targets. No new physical entities or free constants are introduced to force the answers.

free parameters (3)
  • MCMC burn-in steps = 30000 (primary experiments)
    Chosen empirically (~15k–30k) from Hamming-distance and weight-imbalance diagnostics on n=9–10; not fitted to D(n) targets but affects bias if too short for higher n.
  • MIN_SPLIT_COUNT (filter tree) = 131072000
    Empirical threshold (131072000) stopping binary-tree pre-filter splits for pair matching; affects runtime, not the estimator definition.
  • Reference-subset / experiment sample sizes N and K = varies by n
    Chosen for hit-rate and S.E. targets (e.g. 1e8 samples × 40000 experiments for D(10) 1-layer); engineering, not free constants in a model of D(n).
axioms (5)
  • standard math Any MBF in n+1 variables splits uniquely into a comparable pair of n-variable MBFs (Lemma 1 / Corollary 2), so p_match,n = D(n+1)/D(n)^2.
    Standard recursive structure of monotone Boolean functions; used for pair-matching estimator (Theorem 3).
  • standard math Simple random walk on the undirected MBF graph has stationary distribution π(f) ∝ deg(f); degree correction 1/deg(f) recovers uniform averages.
    Standard Markov-chain fact (Levin–Peres–Wilmer cited); applied throughout MCMC sampling and layer branching.
  • standard math The set of 1-layer MBFs free only on a middle layer k has cardinality exactly 2^{C(n,k)}.
    Inputs of equal weight are pairwise incomparable; used as closed-form |S| for reference-subset estimators (Section 4.3).
  • domain assumption After sufficient burn-in, MCMC samples on G(n) for n=10–15 are close enough to stationarity that degree-corrected layer ratios estimate true branching factors within the reported S.E.
    Mixing is diagnosed carefully for n=9–10 but only extrapolated for n≥11; this is the main domain assumption for D(13)–D(15).
  • ad hoc to paper Relative discrepancy at the middle layer between upward and downward weight-layer propagations is a usable proxy for the standard error of the total D(n) estimate.
    Authors take relative error of the most populated layer as S.E. proxy (Section 4.4.1); reasonable but not a theorem.
invented entities (1)
  • Weight Layer Branching estimator no independent evidence
    purpose: Propagate estimated left/right degree ratios from layer 0 and layer 2^n to the middle to reconstruct all layer sizes and thus D(n).
    Methodological construct, not a physical entity; independent evidence is multi-method agreement for n≤12 and midpoint discrepancy diagnostics. independent_evidence left false because the estimator is defined by the paper's procedure rather than an external observable.

pith-pipeline@v1.1.0-grok45 · 19931 in / 3710 out tokens · 37398 ms · 2026-07-10T07:20:04.878825+00:00 · methodology

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read the original abstract

We provide highly accurate estimations of the 10th through 15th Dedekind Numbers, to a precision of 4 digits for $D(10)$, to 2 digits for $D(15)$. These estimates were obtained using three methods, including pair matching on large quantities of 9-dimensional monotone Boolean functions for $D(10)$, Reference Subsets for $D(10)$, $D(11)$, and $D(12)$. And our best method "Weight Layer Branching" which provided accurate estimates for all $D(10)$ through $D(15)$, strongly improving on the previous best known estimates by Korshunov and Tian-Shun Chen et al. arXiv:2606.09795

Figures

Figures reproduced from arXiv: 2607.08446 by Alex Fihman, Christian Plessl, Lennart Van Hirtum.

Figure 1
Figure 1. Figure 1: Degree-corrected weight distribution of MBF9s from MCMC at 5,000 and 30,000 burn-in steps, [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Both figures depict the relative number of MBFs in each layer. Horizontal axis is “progress from [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

discussion (0)

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Reference graph

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