The Capacity Regulator of Absolute Frame Theory III: Exponential Clustering in the Physical Window, Chessboard Estimates, and the Collar Theorem for the Geometric Sector

The first two papers of this series established exact reflection positivity for the capacity-regulated sectors of Absolute Frame Theory (AFT) at finite volume, subsequential infinite-volume states carrying the Osterwalder--Schrader data short of clustering, the existence of the Nambu--Goto-reweighted regulated measure, and the analytic closure of the matrix-mixture route to geometric reflection positivity. This third paper completes the decoupling triad in its physical window and resolves the geometric positivity question in both directions. First, a cluster expansion around the free massive lattice field, with activities controlled by the derived damping and by the routing scale, converges below an explicit threshold: the thermodynamic limit is then unique without subsequences, correlations cluster exponentially, the connected channel correlators are nontrivial, and the full set of Osterwalder--Schrader properties---including clustering and a transfer spectral gap---holds for the regulated scalar sector; the routing component of the expansion runs at the inverse coherence bound 1/N₂ₑ₈ₓ=/c, and the saturation point is named open. Second, the multi-plane reflection positivity of the torus venue yields chessboard estimates by dyadic multi-reflection, and with them the exponential suppression of multi-block super-capacity regions with no independence assumption: the cell-model bound of the companion papers becomes a theorem of the lattice measure itself. Third, a four-configuration witness proves that the raw Nambu--Goto crossing weight is not reflection-positive: folded cells of zero area escape suppression, and the kink of the area element on the rank-deficient locus produces an explicit negative Gram eigenvalue e^-2TH-1; the obstruction is discretization-dependent but robust, and persists for the physical crossing kernels at order-one geometric tension. Fourth, the collar theorem turns this obstruction into structure: with the crossing layer excised at the reflection plane---a collar one cell wide at the action-quantization envelope---the geometric reweighting is exactly reflection-positive; on the Hilbert space already constructed, every geometrically loaded slab word with clean ends or a clean center is positive by elementary palindrome algebra, whence the geometric Euclidean dynamics exists as positive self-adjoint operators with no reflection-positive geometric measure required; and the composition defect of these operators, localized at composition collars and linear in the geometric tension, is identified as the modular mark that the programme assigns to the saturated sector, now localized at one layer of physical resolution. The remaining frontier is stated in prioritized form.