Certificate-Tower Spectral Sequences for Linearized Identity-Eliminator Obstruction Towers

This is Paper 18 in the 20 Paper PHHT Series This paper constructs certificate-tower spectral sequences for linearized identity-eliminator obstruction towers in identity-eliminator-conserved bilateral homotopy type theory. The certificate tower records recursive lifting problems; it is not itself a filtered complex. An ordinary exact-couple spectral sequence appears only after adding a branchwise filtered cochain realization of the linearized abelian tower. For a chosen lower branch, the stage-r associated graded piece of the filtered realization is the relative twisted obstruction complex C^•Rel(K,L{η_<r};A_r). For a singleton retained glut (p,q), this filtered realization determines a branchwise spectral sequence E_a^{u,v}(p,q;η). The stage-r obstruction is the invariant image of the exact-couple transgression: π_r(d_r(ξ^{(r)}{η<r})) = [c_r(p,q;η_<r)]Indet ∈ 𝒬_r(p,q;η<r). Lower-filler variation is controlled by based transport and affine coinvariant obstruction targets 𝒬_r. Certification through retained dimension n is equivalent, under the finite branchwise spectral certification package, to a recursively compatible choice of lower certificates for which all invariant obstruction values vanish through dimension n. The required package includes the boundary law, extension-completeness, realization-complete indeterminacy, null-reflection, transgression compatibility, and indeterminacy-compatible page comparison. For a finite retained store (p,R), the protected retained-family certificate remains the guarded identity-elimination gate: FamCert≤ₙ(p,R) = (∏_{q∈R} Cert≤ₙ(p,q)) × StoreCompat≤ₙ(p,R). Spectral, detected, tower, quotient, and aggregate presentations enter this gate only through explicit preserving, reflecting, and comparison-complete data that preserve the StoreCompat factor. The paper also separates finite-stage certification from full retained certification. Full certification is inverse-limit or homotopy-inverse-limit certification, not merely finite solvability at each bounded stage. In affine abelian towers, the remaining obstruction after finite solvability is the canonical lim¹ class of the finite certificate torsor tower. Filtered morphisms of realized towers preserve exact couples, page transgressions, and invariant obstruction values. Substitution, coefficient-preserving equivalence, type-former structure, cellular–cubical comparison, and obstruction-gated univalence are treated as instances of this functoriality when their declared spectral comparison packages are complete. This paper supplies the finite-to-full gateway for the series. It imports the operation, finite detected, and coefficient layers of Papers XV–XVII and provides the compatible-thread criterion used by the later completion, localization, and synthesis papers.