Worked Cellular Models for Identity-Eliminator Obstruction Calculus

This is Paper 8 in the 20 Paper PHHT Series This paper computes finite detected cellular witnesses for the obstruction hierarchy of identity-eliminator-conserved n-truncated graded types. Each witness is a finite reduced translation test presentation with a declared coefficient system, relative obstruction target, admissible repair map, and residual functional. Polyhedral residuals are evaluated in relative groups Hᵈⁱᵐ P(P,∂P;L_P), so the top obstruction of a disk-like coherence polyhedron is measured before extension across the interior cell. Its finite detected value therefore lives in the relative target prior to passage to the absolute cohomology of the filled disk. The main computational mechanism is an affine value-set calculation. If V_P is the entire admissible repair-variation space and Θ_P(b₀+v) = Θ₀ + λ_P(v), then the associated identity-eliminator operation has value set ⟨P⟩_IdElim = Θ₀ + λ_P(V_P). Over 𝔽₂, the all-ones boundary-to-top functional annihilates every even-support repair variation. Consequently, whenever the full admissible repair-variation space lies in this kernel, any nonzero base residual survives all admissible repairs. The worked presentations compute several finite detected witnesses: a nonzero first reduced class on a one-loop quotient; a nonzero exact primary cocycle with nonunique primitive on a filled triangle; a surviving secondary exchange residual on a square; a surviving Yang–Baxter residual on a hexagon after all selected elementary braid repairs vanish; a nonzero cubic interior datum after all square faces fill; and a surviving associahedral pentagon residual. These models prove finite detected separations between primary vanishing, primitive data, selected secondary coherence, selected polyhedral coherence, cubical interior data, associahedral coherence, and filling depth. They show that the ordinary primary obstruction tower does not determine the primitive torsors, secondary residuals, higher polyhedral residuals, or depth data recorded by the finite cellular complexity package 𝔠≤ₙ^cell = (𝔠_prim,𝔠_sec,𝔠_poly). All results are finite detected statements in declared cellular or cubical detectors. They are not retained-family certificates or full inverse-limit certificates by themselves. Retained-family certification and full certification require the reflecting comparison and compatible-thread hypotheses supplied by later certificate layers.