Worked Detected Obstruction Calculus for Identity-Eliminator-Conserved Bilateral HoTT

This is Paper 16 in the 20 Paper PHHT Series This paper develops a worked finite detected obstruction calculus for identity-eliminator-conserved bilateral homotopy type theory. For a retained identity glut (p,R), the abstract obstruction tower assigns recursively indexed pointed obstructions o_k(p,R;η_<k) ∈ Ob_k(p,R;η_<k), where each stage is defined only after the lower filler package η_<k has been chosen. The paper gives finite detected presentations of this tower through a retained dimension n. These include strict additive cellular detectors, fixed-boundary relative cellular detectors, and cubical box-filling detectors. In the relative cellular regime, the basic one-relative-top-cell calculation H^k(K,L;A) ≅ A_σ controls explicit interval, triangle, tetrahedron, and disk computations. The main structural theorem identifies finite detected solvability with finite tower certification under extension-completeness and detector-completeness. A detected solution d : DetSol^𝒟_≤ₙ(p,R) is equivalent to a tower certificate TowerCert_≤ₙ(p,R) when the detector is complete through dimension n and the comparison data preserve lower fillers, retained-store restrictions, indeterminacy transport, singleton inclusions, truncation, and StoreCompat. Algorithmic effectivity is treated as a separate witness-extraction condition. When the finite stage equations and lower-filler searches are decidable, the detected calculus gives a terminating procedure that either produces a detected witness or returns the first failed obstruction stage. Aggregate realization-completeness and the complete certificate comparison package are also separate promotion hypotheses. They carry tower certificates into the protected retained family-certificate gate FamCert_≤ₙ(p,R⁺), where R⁺ is the interaction-closed retained store. The comparison target descends to FamCert_≤ₙ(p,R) only under interaction-closure or a supplied protected non-interaction equivalence, including descent of the StoreCompat component. The result is explicitly finite-stage. A finite detected certificate through dimension n does not by itself give full retained certification. Full certification requires a compatible inverse-limit or homotopy-inverse-limit thread satisfying the stated tower-fibrancy, compactness, convergence, or limit-coherence hypotheses. The worked examples exhibit primary, secondary, and tertiary obstructions, detector-level composite cancellation, dependent-sum and dependent-product coupling obstructions, naturality failure, and the identity-type dimension shift. The paper thereby supplies the concrete finite detected computation layer connecting the recursive obstruction-operation algebra to later finite-to-full, coefficient, spectral, and completion results in the series.