Refined Spectral Shadows and Higher Differential Structure of Identity-Eliminator-Conserved Graded Types

Paper 4 in the 20 Paper PHHT series. This paper constructs the filtered obstruction refinement of the obstruction tower attached to identity-eliminator-conserved graded types. For a fixed obstruction layer m, the paper studies the filtered cochain complex Cₘ^• = C^•(B𝒢≤n;𝒜ₘ), with distinguished obstruction representative cₘ ∈ Cₘᵐ and obstruction class ωₘ = [cₘ] ∈ Hᵐ(Cₘ^•). A finite filtration preserved by the twisted coboundary determines a filtered shadow spectral sequence Eᵣᵖ,ᵠ(m) ⇒ grᵖHᵖ⁺ᵠ(Cₘ^•). The page differential dᵣ is interpreted as an internal filtered obstruction refinement within the fixed layer m. If x ∈ FᵖCᵖ⁺ᵠ represents a page-r class and δx ∈ Fᵖ⁺ʳCᵖ⁺ᵠ⁺¹, then dᵣ[x] = [δx] ∈ Eᵣᵖ⁺ʳ,ᵠ⁻ʳ⁺¹(m), and this class measures the obstruction to replacing x, within its page-r class, by a representative whose coboundary lies one filtration step deeper. For split filtrations, the intrinsic differential admits an explicit computational representative dᵣ[x₀] = [∑_{i+j=r, 0≤j≤r−1} δₘ,ᵢxⱼ], where x₀,…,xᵣ₋₁ solve the lower correction equations. These formulas express the higher differentials as correction obstructions for leading filtered representatives. The paper defines filtered shadow profiles recording filtered cochain complexes, associated graded pieces, leading obstruction representatives, spectral sequence differentials, death pages, filtered obstruction persistence intervals, comparison maps, terminal associated-graded components, and terminal assembly data. The enhanced marked profile reconstructs the distinguished obstruction class ωₘ from its terminal associated-graded pieces together with assembly data. The spectral shadow tower Sh≤n refines the ordinary obstruction tower Ω≤n, and the forgetful passage Sh≤n → Ω≤n has nontrivial marked-filtered fibres. In particular, finite coboundary-nondecreasing cellular filtrations may have ωₘ = 0 while retaining nonterminal spectral ghosts or positive marked primitive depth. Thus ordinary obstruction-theoretic vanishing does not determine the filtered correction structure of the distinguished representatives. Over finite-support finite-dimensional pages, the filtered obstruction complexity vector 𝒞ₘ^ω = (gₘ^ω, Rₘ^ω, pₘ^ω, tₘ^ω) detects ordinary tower vanishing exactly through its terminal ranks tₘ^ω, while retaining filtered obstruction information discarded by Ω≤n. The constructions remain obstruction-refinement data; retained certificate consequences and guarded identity-eliminator admissibility require later detected-to-retained, finite-to-retained, or aggregate-nullity comparison layers.