Secondary Obstruction Operations for Identity-Eliminator-Conserved n-Truncated Graded Types

This is Paper 5 in the 20 Paper PHHT Series This paper constructs the secondary obstruction calculus for identity-eliminator-conserved n-truncated graded types. Let X=B𝒢≤n be the reduced cellular complex of a presented n-truncated graded obstruction type. After selected-primary vanishing, the pulled-back primary class ω₁ˣ = f∗ω₁ᵘⁿⁱᵛ is nullified, and each distinguished primary cocycle cₘ ∈ Zᵐ(X;𝒜ₘ) comes equipped with primitive data bₘ₋₁ ∈ Cᵐ⁻¹(X;𝒜ₘ), δ𝒜ₘbₘ₋₁ = cₘ. The primitive choices form torsors over Zᵐ⁻¹(X;𝒜ₘ). The central question is whether these primary primitives can be chosen compatibly. For admissible primitive pairs, a secondary interaction datum Qᵢⱼ assigns a closed cochain representative sᵢⱼ,Q(bᵢ₋₁,bⱼ₋₁) = Qᵢⱼ(bᵢ₋₁,bⱼ₋₁) ∈ Cⁱ⁺ʲ⁻¹(X;ℬᵢⱼ). The resulting secondary operation is not a strict cohomology class but a quotient-valued operation modulo the displayed indeterminacy subgroup: ⟨ωᵢˣ,ωⱼˣ⟩Q,ℐ ∈ Hⁱ⁺ʲ⁻¹(X;ℬᵢⱼ)/ℐQ. The paper proves that strict binary cup-boundary expressions give zero cohomology classes, so nonzero secondary operations require interaction data not reducible to strict boundary form. For a finite strict interaction family 𝒬, the simultaneous secondary compatibility problem is encoded by the global secondary obstruction map Φ𝒬:𝒫(c•) → ∏Q∈𝒬 H|Q|(X;ℬQ). When 𝒬 is cohomologically affine-linear, Φ𝒬 is an affine map of primitive torsors with linear part L𝒬. The obstruction to global secondary nullification is then the basepoint-independent cokernel class 𝔰𝒬 = Φ𝒬(b⁰) ∈ coker L𝒬. In the cohomological filling calculus, and more generally after passage to any null-reflecting realization quotient, global secondary nullification exists exactly when this cokernel class vanishes. The paper also separates individual secondary quotient-nullity from simultaneous quotient-nullity. Individual nullity is coordinatewise vanishing of singleton secondary obstruction classes, while the precise obstruction to assembling those individual nullifications into a global compatible nullification is the kernel of the natural comparison coker L𝒬 → ∏Q∈𝒬 coker LQ. A finite strict cochain model realizes this kernel obstruction, showing that separate representative-level secondary nullifiability need not assemble into global secondary identity-eliminator conservation in the displayed quotient calculus. The paper concludes by situating the secondary calculus relative to the primary obstruction tower and the filtered-shadow refinement: primary obstruction classes detect selected cohomological obstruction, filtered shadows refine individual obstruction layers, and secondary operations measure compatibility among nullifying primitives across layers. A typed threefold Massey datum provides the first tertiary operation, replacing strict associativity with an explicit associator cochain in the homotopy-coherent setting.