Coefficient Systems and Indeterminacy for Identity-Eliminator Obstruction Operations

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 A_k : 𝒢_IdElim(𝒞) → Ab. A detected cellular presentation pulls this system back along a cellular indexing functor to a flat cellular local system A_{k,η} = A_k ∘ γ_{k,η} on a finite identity-coherence complex K. The associated twisted coboundary satisfies d_A² = 0. When the detected boundary term is supported on the fixed lower relative boundary L_{η_<k}, the admissible projection of the detected obstruction representative is a relative twisted cocycle and defines a class [c_k(p,q;η_<k)]Rel ∈ H^k_Rel(K,L{η_<k};A_{k,η}). 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 H^k_Rel(K,L_{η_<k};A_{k,η}) / I_k(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.