Twisted Higher Obstruction Towers for Identity-Eliminator-Conserved n-Truncated Graded Types

This is Paper 2 in the 20 Paper PHHT Series. This paper constructs the twisted higher obstruction tower for identity-eliminator-conserved n-truncated graded types. Starting from a selected reduced presentation f:X→B𝒢≤n, the primary obstruction is the pullback parity class ω₁ˣ = f*ω₁ᵘⁿⁱᵛ = [χ∘disp_X] ∈ H¹(X; 𝔽₂ᵈ), so primary vanishing is always interpreted relative to the selected presentation, not as universal vanishing on B𝒢≤n. For each k≥2, a partial identity-eliminator-conserved solution through the (k−1)-skeleton, a cellular local coefficient system 𝒜_k, transported identity-eliminator defect data Def_Id⁽ᵏ⁾, boundary coherence, and a boundary-compatible filling condition determine a twisted cellular cochain c_k ∈ Cᵏ(X;𝒜_k). Boundary coherence makes c_k closed, and lower-dimensional coherence modifications change c_k by a twisted coboundary. Hence ω_k = [c_k] ∈ Hᵏ(X;𝒜_k) is the k-th obstruction class. Vanishing of ω_k is necessary for extension, and sufficient under the stated boundary-compatible filling hypotheses. The formal choices killing c_k form the corresponding torsor before passage to semantic fillers. The paper develops the tower in absolute, relative, skeletal, gluing, naturality, finite detected cochain-calculation, and minimal-cellular-model forms. It also separates representative-level vanishing, class-level vanishing, nullity data, and realized fillers. Naturality is class-level unless representative and filler transport data are supplied, and minimality is relative to cellular obstruction data realizing the declared classes. The non-collapse theory is given in two forms. Axiomatic cellular non-collapse shows that prescribed closed representatives yield towers with all lower stages zero and a nonzero top class. Semantic non-collapse is realized through central-extension and Postnikov realization packages: the Heisenberg central extension realizes the dimension-two class, while for every k≥3 a connected Postnikov stage with π₁=ℤᵏ, πₖ₋₁=ℤ, and primitive k-invariant [u_k]∈Hᵏ(ℤᵏ;ℤ) realizes ω₁=⋯=ωₖ₋₁=0, ω_k=1∈Hᵏ(Tᵏ;ℤ). Thus primary parity conservation is only the first layer of identity-eliminator conservation; the full theory is a genuinely higher obstruction tower.