AbstractWe construct the extended directive quasi-metric geometry associated to obstructiondescent in Chromatic Obstruction Sheaf Theory. The construction begins from the rela-tional descent environment of COST I and the fundamental stable obstruction packageof COST II. Thus a small finitely complete relational site (C, J) determines the hyper-complete relational ∞-toposX := ShvhypS (C, J),together with the fundamental unstable obstruction carrier Ofund ∈ X∗, its stable spectralimageFfund := Σ∞Ofund ∈ Sp(X ),and the associated stable coefficient sheaves Afund,n := πn(Ffund). For connective stableobstruction data, local lifting failures are measured by stable Postnikov obstruction classesostfund,n(s) ∈ Hn+1(U ; Afund,n).The main point of this paper is that such obstruction classes do not by themselves de-termine functions, gradients, lengths, or metrics. To pass from lifting failure to geometry,we introduce categorical admissible invariant density realizations. These send obstructionrepresentatives, compatibly with refinement and representative equivalence, to invariantobstruction-density classes. After sheafification, these classes form obstruction-densitysheaves over the relational site. Regular analytic representatives of such density classes,together with admissible pairings and directive-admissible curve classes, then producedescent vector fields and directive path functionals.Given a regular local representative ρ of a total obstruction-density class and anadmissible positive-definite pairing g, the obstruction descent vector field isVobs[ρ] := −∇ρ.Along regular descent trajectories, the obstruction density is nonincreasing:ddtρ(γ(t)) = −∥∇ρ(γ(t))∥2g ≤ 0.The associated directive path functional penalizes motion in directions of increasing ob-struction density, and its infimum defines an extended directive quasi-metric Dobs. Thiscomparison function is nonnegative, vanishes on the diagonal, and satisfies the directedtriangle inequality, but it need not be symmetric and may take the value +∞.We prove that the resulting directive geometry is compatible with restriction, refine-ment, sheafification, and hyperdescent, provided the density realization, regular represen-tatives, pairings, and curve classes satisfy the stated compatibility hypotheses. We thenspecialize the construction to the fundamental chromatic obstruction package, includingheight-filtered, monochromatic, and fracture-compatible density data. Finally, we estab-lish layered base-change and Morita-invariance results: the stable obstruction packageis inherited from COST II, invariant density classes are transported by Morita-naturalrealization, and the extended directive quasi-metric is invariant under additional regularanalytic and curve-class compatibility hypotheses.Thus COST III supplies the first intrinsic geometry of obstruction release in COST.It does not impose a background metric, causal structure, physical action, or externaldynamical law. Instead, it derives a directed comparison geometry from stable obstructionclasses only after admissible density realization and regular directive data have beensupplied. This provides the geometric bridge from the foundational obstruction theoryof COST I–II to the derived moduli and cotangent constructions developed in COST IV.