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∘dispX ∈ 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 𝒜ₖ, transported identity-eliminator defect data DefId⁽ᵏ⁾, boundary coherence, and a boundary-compatible filling condition determine a twisted cellular cochain cₖ ∈ Cᵏ (X;𝒜ₖ). Boundary coherence makes cₖ closed, and lower-dimensional coherence modifications change cₖ by a twisted coboundary. Hence ωₖ = cₖ ∈ Hᵏ (X;𝒜ₖ) is the k-th obstruction class. Vanishing of ωₖ is necessary for extension, and sufficient under the stated boundary-compatible filling hypotheses. The formal choices killing cₖ 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 uₖ∈Hᵏ (ℤᵏ;ℤ) realizes ω₁=⋯=ωₖ₋₁=0, ωₖ=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.