Finite Detected Cellular Algorithms for Polyhedral Identity-Eliminator Obstruction Calculus

This is Paper 7 in the 20 Paper PHHT Series. This paper gives finite detected cellular algorithms for the obstruction calculus associated to identity-eliminator-conserved n-truncated graded types. The input is a validated finite detected obstruction package FinObsPkg≤ₙ(X,D), consisting of a finite regular cell complex, flat transported local-coefficient systems, distinguished primary cocycles, selected secondary and polyhedral residual rules, correction maps, variation spaces, and quotient presentations, all given by explicit finite matrices. In this finite detector regime, every pointwise obstruction-vanishing problem has the same linear form. For a cocycle v∈ker D⁺, nullity modulo correction and allowed variation is equivalent to the membership condition v ∈ im(D⁻ V). This criterion specializes uniformly to primary obstruction vanishing, primitive search, linearized secondary vanishing, quotient-presented nullity, and selected polyhedral residual vanishing. The paper distinguishes pointwise finite detected nullity from global finite detected coherence. Pointwise nullity tests the selected layers, operations, and polyhedra separately. Global coherence requires one simultaneous choice of primary primitives, secondary target corrections, and polyhedral correction cochains. Under the joint-affine hypothesis on the selected secondary and polyhedral residual rules, these requirements assemble into a single finite affine system Aξ = c. Over a field, finite detected global coherence through dimension n is therefore equivalent to the rank condition rank A = rank(A c). The global condition implies all selected pointwise obstruction indicators vanish, but the converse is strict: separately vanishing pointwise tests may require incompatible primitive or filler choices. The algorithms return exactly one of three finite detected outcomes: ValidationFailed, FiniteDetectedSolved, FiniteDetectedObstructed. Successful outputs are finite detected nullity, finite detected obstruction, and finite computational filler data in the declared detector. These outputs are not automatically retained, semantic, torsor-level, orbit-level, or full inverse-limit certificates. They become statements about retained targets or full retained certification only through the corresponding reflecting, complete, or compatible-thread comparison hypotheses supplied later in the series. This paper supplies the algorithmic layer of the series: validated finite detector packages, matrix normal forms for local-coefficient cellular cohomology, finite membership tests, primitive-depth computation, linearized secondary-operation computation, selected polyhedral residual computation, global affine coherence tests, finite witness construction, and finite detected obstruction classification.