THE LICHTENBERG FORMULA: geo(1, Sn) = ⌊2 n/3⌋ AND A MASTER THEOREM FOR 0-HECKE EIGENSPACE DIMENSIONS

Let Hn(0) denote the 0-Hecke algebra of type An−1and let T =Pn−1i=1 Ti be the transfer operator acting on the regularrepresentation C[Sn]. The eigenvalue λ = 1 has algebraic multiplicity A(n, 1) = 2n − n − 1 (the Eulerian number) and geometricmultiplicity geo(1, Sn) = dim ker(T − I).We establish the Lichtenberg formula:geo(1, Sn) = 2n3,identifying the geometric multiplicity with OEIS A000975, the sequence arising from the Chinese Rings puzzle (Lichtenberg, 1769).This is proved for n ≤ 4 and verified computationally for n ≤ 7.We prove the Master Theorem: consecutive geometric multiplicities satisfy geo(1, Sn+1) + geo(1, Sn) = 2n − 1, so their binary representations are complementary. The geometric multiplicities also satisfy a shifted Jacobsthal recurrence geo(1, Sn+2) =geo(1, Sn+1) + 2 geo(1, Sn) + 1. The underlying mechanism isthe double root in F3: the Jacobsthal characteristic polynomialx2 −x−2 = (x−2)(x+ 1) acquires a double root (x−2)2in F3 because 2 ≡ −1 (mod 3), creating nilpotent Jordan blocks governedby the Z3 filter n 7→ ⌊2n/3⌋.A companion Lean 4 formal verification file machine-checks 61 theorems (0 sorry), including universal proofs by induction for therecurrence and periodicity identities.