This is Paper 6 in the 20 Paper PHHT Series This paper constructs a polyhedral obstruction calculus for higher identity-eliminator coherence in identity-eliminator-conserved n-truncated graded types. For the reduced translation complex X = B𝒢≤n, compatibility data are indexed by finite face-closed selected families of admissible coherence polyhedra P→X, including exchange squares, Yang–Baxter hexagons, cubical interchange cells, permutohedra, associahedra, and mixed cells. The input for each polyhedral theorem is a package PolyPkg≤n (P), consisting of admissibility data, oriented incidence data, selected-primary nullification, lower-stage solutions, compatible boundary corrections, transported twisted coefficient systems, and the Paper V indeterminacy data used when secondary operations enter. A regular selected incidence system with flat transported defect coefficients determines a cellular cochain complex (C•𝔓 (X;𝓛𝔓^ (r), ρ), δρ). Given dependency-complete lower filling data, each selected polyhedron P determines a relative residual cochain rP ∈ Cᵐ (P, ∂P;ιP∗𝓛P^ρ), its residual obstruction class ωP = rP ∈ Hᵐ (P, ∂P;ιP∗𝓛P^ρ), and, when secondary data enter, a quotient residual class modulo the relevant indeterminacy subgroup. The relative theorem states that ωP is the obstruction to extending a fixed filled boundary across P. Any extension forces ωP=0; conversely, ωP=0 together with a boundary-compatible realized filler gives an extension. The absolute theorem is obtained only after all proper faces and dependency predecessors have already been solved compatibly. The construction separates cellular degree from operation stage. The cohomological degree of a residual is dim P, while the operation stage is the height of P in the fill-dependency order. Thus, for example, a Yang–Baxter cell may have cellular degree 2 while functioning as a tertiary operation because its residual is defined only after lower exchange data have been filled. Polyhedral coherent identity-eliminator conservation is defined as finite coherence solvedness in this selected incidence calculus. Retained identity-eliminator certification is not automatic; it enters only through later retained-target comparison theorems. In finite linear correction systems over a principal ideal domain, realized polyhedral residuals become cokernel classes computable from finite incidence, transport, and correction matrices. This paper therefore supplies the polyhedral higher-coherence layer of the series: admissible incidence systems, residual cochains, residual obstruction classes, quotient residuals modulo indeterminacy, and finite polyhedral fillers for use in later cellular algorithms, finite worked models, coefficient-system constructions, and aggregate comparison frameworks.
David Betzer (Mon,) studied this question.