FOUNDATION — Volume I — The Lattice Dual Triangularity, Signed Transpose Similarity, and Isospectrality in the 0-Hecke Monoid

Let (W, S) be a finite Coxeter group with K = |S| generators. The 0-Hecke monoid H₀(W) acts on the group algebra ℂ[W] via idempotent operators πₛ satisfying πₛ² = πₛ and the braid relations. We study two natural operators on ℂ[W]: the transfer operator T = Σ πₛ and the descent operator D = Σ(πₛ − sₛ), where sₛ denotes the simple reflection. We prove two main results. Theorem A (Dual Triangularity): in any linear extension of the descending Bruhat order (with w₀ first and e last), T is upper triangular and D is lower triangular with the same diagonal entry des(w) at each position w ∈ W, yielding χ_T(λ) = χ_D(λ) = Π(w∈W)(λ − des(w)). Theorem B (Signed Transpose Similarity): the identity D = J Tᵀ J holds, where J = diag((−1)^{ℓ(w)}) is the length-parity matrix. Since every matrix over a field is similar to its transpose (Taussky–Zassenhaus), this implies T ∼ D, establishing that T and D share not only their characteristic polynomial but their full Jordan normal form. We verify the identity D = J Tᵀ J computationally for types A₂–A₆ and B₂–B₃ and explore the resulting Jordan structure. The eigenvalue multiplicities recover the Eulerian numbers (type A) and type-B Eulerian numbers (OEIS A060187). The geometric multiplicity sequence at λ = 1 coincides with the Lichtenberg sequence (OEIS A000975). We exhibit a third isospectrality mechanism on the Boolean lattice 𝒫([n]), where conjugation by the complement involution C replaces the signed transpose. We conclude with open questions on q-deformation, the Norton product connection, and classification of isospectral operator pairs on graded posets. Keywords: 0-Hecke monoid, Coxeter group, transfer operator, descent operator, isospectrality, signed transpose, Jordan normal form, Eulerian numbers, Lichtenberg sequence, Boolean lattice, dual triangularity. MSC 2020: 20F55, 05E16, 15A21, 20M30, 06A07.