Poset Stratification and Associated Graded Structures of the Characteristic Spectrum of Profinite Eventually Periodic Sequences

Within the framework of profinitely eventually periodic sequences (RFEP) established in the first paper, we study the global poset stratification and the associated graded structures induced by the characteristic spectrum map. We normalize the spectrum value domain as V = N0 × N>0, where the first coordinate is ordered by the usual order and the second by divisibility. We regard the characteristic spectrum of an RFEP sequence as a whole as a spectrum object Λ_A = (N_A, T_A), rather than fixing any index layer. By the divisibility monotonicity already proved in the first paper, the characteristic spectrum map lands in the space of spectrum functions ordered pointwise by the spectrum coordinates. For every spectrum upper bound σ ∈ S↑, we define the down-closed filtration piece F_⪯σ = {A ∈ P̂ : Λ_A ⪯ σ}. We prove that each down-closed filtration piece F_⪯σ is a unital Ẑ-subalgebra of P̂ (closed under termwise addition, Ẑ-scalar multiplication, and pointwise multiplication), so that the family {F_⪯σ} forms an exhaustive algebraic filtration indexed by the spectrum poset. We further construct the strict lower filtration and the associated graded object, and record the algebraic filtration relations on it; when the standard well-definedness condition for the multiplication on the associated graded quotients holds, the associated graded object becomes a poset-graded algebra indexed by S↑. Finally we show that the nilpotent–pure-periodic decomposition separates the two coordinates in this stratification into the pure-periodic projection and the nilpotent projection.