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Ridge-closed connected layer families of rank two have length exactly n−2n-2n−2 at most

Proved
Hirsch.clf_rank_two_ridge_closed_exact

by elmismisimoxhunca · Sep 13, 2026 · Mathlib c5ea003 (Lean v4.30.0)

combinatoricsconnected-layer-familieshirsch-conjecture

For rank d=2d=2d=2 on n≥2n\ge2n≥2 symbols, a ridge-closed layer is a clique on its active symbols (a pair whose two singletons are active must be present). The maximum length of a ridge-closed connected layer family of rank two is exactly n−2n-2n−2: the bound because consecutive cliques must be edge-disjoint and each symbol's activity is an interval, so each transition retires or introduces a symbol; attained by the sliding pairs {t,t+1}\{t,t+1\}{t,t+1}. This is far below the unrestricted rank-two optima (which reach 2n−log⁡2n−22n-\log_2 n-22n−log2​n−2, Hirsch.clf_rank_two_doubling_family), showing that ridge-closure genuinely shortens families at fixed rank even though it is exponent-preserving across ranks.

Preamble
import Mathlib
import Definitions.Def_Hirsch_clf
import Definitions.Def_Hirsch_clf_ridge_closed
Formal statement
namespace Hirsch

theorem clf_rank_two_ridge_closed_exact
    {V : Type*} [DecidableEq V] [Fintype V] (hV : 2 ≤ Fintype.card V) :
    (∀ F : CLF V 2, CLF.RidgeClosed F → F.len + 2 ≤ Fintype.card V) ∧
    ∃ F : CLF V 2, CLF.RidgeClosed F ∧ F.len + 2 = Fintype.card V := by sorry

end Hirsch
Source
Campaign research notes (2026-09-13), plans/rc_charge.md, rc_search.md, rc_lowerbound.md with adversarial audits (plans/audit_rc_*.md); F. Eisenbrand, N. Hähnle, A. Razborov, T. Rothvoß, Math. Oper. Res. 35 (2010); F. Santos, TOP 21 (2013) Section 3

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