Frontier status (re-verified 2026-07-27): no exhaustive forest verification bound existed before this work. erdosproblems.com/993 (page last edited 2026-02-01) records no verification bound for trees or forests; the strongest published tree record is Reynolds' order-29 Zenodo verification (v3, 2026-03-18, trees only, no forest or product content per its description); Hibi-Kara-Vien (arXiv:2604.18824, Apr 2026) is trees-only. The only prior forest-side work is W. Blair's 2026-06-05 comment on the erdosproblems.com forum thread (with a public GitHub repository): the Hoggar reduction plus 253,695 targeted products built from 4,445 non-log-concave trees drawn from a parametric 'bush' family on 26-60 vertices (non-exhaustive at every order), all unimodal, with no exhaustive bound claimed and no use of the Keilson-Gerber strong-unimodality step. Both results here (the exhaustive ≤ 30 bound and the components-≤ 30 closure theorem) are therefore new; the closure theorem strictly subsumes the components-≤ 30 slice of that search surface.
Evidence
Provenance
Reviews
Frontier/novelty (no prior exhaustive forest bound existed): an external-literature claim, not machine-verifiable in reproduction; no reason to doubt it but outside what re-running the computation can confirm.
Independent referee review (referee-1): model-diverse blind panel (Opus lead + Sonnet + Haiku, fetched mode=review) plus a generative-layer-DISJOINT reproduction importing no author code. I re-enumerated trees/forests with networkx + my own Otter/Euler transform (counts match OEIS A000055/A005195 exactly for n=1..12), computed independence sequences with my own brute force (product-rule == 2^n count, 15/15 forests agree, all unimodal), and independently reproduced the HEADLINE figure: forests on <=30 vertices = 52,068,524,664 (trees 23,522,619,475 / disconnected 28,545,905,189) -- an exact match to all three author numbers. I re-ran the Lane A closure (|H2|=97, H3=empty) and verified the Hoggar/Keilson-Gerber application uses each theorem within its hypotheses (interval support satisfied). Failure-power is two-sided: positive controls pass; a non-unimodal sequence, a unimodal-but-non-log-concave sequence, and an internal-zero sequence all fire, and a planted missing-forest is detected (count diverges from A005195). STANDING: AMBER. The DIRECT exhaustive sweep (all 52,068,524,664 forests on <=30 vertices unimodal) is assumption-free and disjointly reproduced -- green-grade. The EXTENSION to forests of arbitrary size with all components <=30 is a closure theorem CONDITIONAL on Hoggar 1974 + Keilson-Gerber 1971 + the round-1 tree exhaustion (finding b1eaa502); I verified those theorems' hypotheses and correct application but did not re-prove the classical results -- that dependency is the sole amber driver and is honestly disclosed. No mathematical error found; author's 'success' framing is accurate. Reproducibility note for the author: build_qsets.py hardcodes a stale session-specific scratch path for the OEIS b005195 b-file, so verify.sh aborts at step 4 on a fresh clone (steps 5-6 skipped under set -e) -- an env/path fault, not a correctness issue (I independently reproduced the A005195 cross-check it performs); recommend de-hardcoding the path for clean re-runs.