Functorial Obstruction Towers, Bridge Realization, and Skeletal Obstruction Packages for Identity-Eliminator-Conserved n-Truncated Graded Types

This is Paper 3 in the 20 Paper PHHT Series and develops the functorial comparison theory for obstruction towers associated to identity-eliminator-conserved n-truncated graded types. A reduced graded obstruction datum consists of a reduced translation complex B𝒢≤n, a ladder of local coefficient systems 𝒜₁,…,𝒜ₙ, and distinguished cellular obstruction cochains cₖ. For cocyclic staged data these cochains define obstruction classes ωₖ ∈ Hᵏ(B𝒢≤n;𝒜ₖ), where ω₁ is the universal first reduced obstruction class represented by the parity-displacement cochain, and for k≥2, ωₖ is the transported identity-eliminator defect class at stage k. The main functorial mechanism is contravariant: a cellular map pulls obstruction classes back, while a reflecting coefficient-ladder map compares the pulled-back target coefficients with the source coefficients. This yields a hierarchy of comparison morphisms. Seed-natural morphisms transport chosen cochain representatives and strict obstruction data; gauge-seed-natural morphisms transport representatives and nullifiers up to a displayed gauge correction; class-natural morphisms transport only cohomology classes and their vanishing; and tower-natural morphisms transport the staged tower only after the lower-stage choices and coefficient systems have been made compatible. Full obstruction invariance is obtained only along tower-equivalence edges carrying two-sided representative, cohomology, and filler comparison data. The paper also introduces comparison-complete obstruction shadows, which retain the cochain complexes, differentials, and distinguished representatives underlying pointed obstruction classes. These shadows explain the information lost when passing to pointed cohomology alone, support rank–nullity and finite-dimensional Hodge decompositions in the linear case, and provide conditional minimal-reduction and classification theorems under explicit realizability, confluence, and shadow-realizability hypotheses. Bridgeable identity-eliminator regimes provide finite local sources for obstruction towers through comparison cocycles on overlap nerves. Under the stated bridge realization hypotheses, they supply the realization packages required by Paper II: the primary obstruction is the selected pullback of the universal first reduced obstruction class, and the second obstruction is realized as a central H²-lifting obstruction, including the twisted local-system case. Finally, the ordinary skeletal filtration is organized as a family of row-q=0 spectral sequences whose E₁-pages are cellular obstruction cochain complexes and whose E₂-pages contain the obstruction classes. Thus the skeletal spectral package organizes the cellular tower without introducing artificial higher transgressions; higher transgression phenomena belong instead to separate semantic or Postnikov filtrations.