The fourth paper of this series left a prioritized frontier whose first three items this paper closes or substantially advances. First, the cell model: the per-cell damping that defines the capacity regulator is shown to be a theorem of Axiom I rather than an input. Three links are derived—weight multiplicativity over independent cells forces an exponential of consumed information by a functional equation; Gaussian extremality of the differential entropy fixes the per-mode information; and the embedding envelope makes that ratio equal to the density over the capacity with no free constant—while the fourth, that the per-cell weight is the exponential of the information consumed, is the Information-Bottleneck clause of the axiom itself. The water-level law is thereby the canonical Bernstein function of the inverse capacity, and the form of the regulator chain of Papers I–II—the exponent and the displacement law—is upgraded from reduced to derived, the capacity scale itself remaining inherited from those papers. Second, the gauge action: the mollified embedding Jacobian is an exactly independent Gaussian matrix, whence the induced metric is Wishart, and a small-ball law for every eigenvalue—each with an exponent equal to half the codimension of the corresponding rank-deficient locus, the derived dimension N=14 grading the whole eigenvalue spectrum—closes, by Hölder with marginals alone and a conditional regression of the Hessian, the first moment of the fully contracted Yang–Mills scalar: the composite gauge field has an integrable action density under the regulated measure in the physical window. Third, volume-uniform clustering: the two-point function of the field channel clusters exponentially at every activity with volume-independent constants, by a conditional killed-walk identity and a Peierls argument over heavy chains; the entire odd sector follows; the t-marginal is shown to be FKG, reducing the even sector to the decay of a single positive order parameter, the conductance pair correlation; a monotone two-sided bound brackets that correlation by an extremal influence function that is finite-volume decidable; and a convergent cluster expansion around the column-product reference establishes its exponential decay for spatial coupling below an in-principle-computable, parameter-uniform threshold. The remaining residue is the strong-spatial-coupling regime, identified as a phase question rather than a technical gap. The frontier is restated accordingly.
Patricio E. Valenzuela (Tue,) studied this question.