This is Paper 17 in the 20 Paper PHHT Series This paper defines the coefficient-system and indeterminacy layer for identity-eliminator obstruction operations in identity-eliminator-conserved bilateral homotopy type theory. For each obstruction stage k, positive identity data determine a coefficient local system Aₖ: 𝒢IdElim (𝒞) → Ab. A detected cellular presentation pulls this system back along a cellular indexing functor to a flat cellular local system A₊, ⏖ = Aₖ ∘ γ₊, ⏖ on a finite identity-coherence complex K. The associated twisted coboundary satisfies dA² = 0. When the detected boundary term is supported on the fixed lower relative boundary L⏖_<₊, the admissible projection of the detected obstruction representative is a relative twisted cocycle and defines a class cₖ (p, q;η_<k) Rel ∈ HᵏRel (K, Lη_<k;A₊, ⏖). The paper separates raw obstruction operations, detected representatives, relative cohomology classes, and invariant quotient classes. Compatible lower-filler variations act on the family of relative obstruction targets, so the invariant obstruction is not a single representative in a fixed group. Depending on the target, it is an orbit object, action groupoid, homotopy quotient, affine coequalizer, or, in the translational abelian case, a quotient HᵏRel (K, L⏖_<₊;A₊, ⏖) / Iₖ (p, q;η_<k). Vanishing modulo indeterminacy is therefore target-qualified nullity in the quotient target, not strict vanishing before quotienting. It extends a certificate only under the displayed null-reflection, realization-completeness, and fibrewise extension-completeness hypotheses. The coefficient formalism also supplies structural obstruction criteria for dependent sums, dependent products, identity types, equivalences, and obstruction-gated univalence. These criteria require the relevant coefficient systems, relative-boundary projections, torsor or quotient targets, and indeterminacy comparison data. The paper gives the retained-coefficient apparatus needed to compare finite detected obstruction calculations with protected retained certificate gates. For a retained-store glut (p, R), complete coefficient-level comparison identifies the StoreCompat-inclusive family certificate FamCert≤ₙ (p, R) with the corresponding detected coefficient package, including singleton certificate factors, coefficient transport, lower-filler transport, relative-boundary transport, indeterminacy transport, and guarded-exposure compatibility. This paper supplies the coefficient and indeterminacy conventions used by the later spectral sequence, protected certificate completion, localization, and synthesis papers in the series.
David Betzer (Tue,) studied this question.