Finite Universe Cosmology, Part III: Finite Spatial Volume from Geometric-Address Capacity

We prove a self-contained global theorem that derives finite physical spatial volume from finite persistent geometric-address capacity in closed physical readout systems. A whole-domain commutative geometric-address algebra generates the realized address quotient while retaining vacuum, background, and interaction sectors and removing duplicate representations. On its regular locus, the canonical address experiment defines a Fisher--Jeffreys measure, and one address-blind common implementation of the complete family determines a persistent worst-case excess log-loss. Under the compact regularity conditions of Clarke--Barron minimax redundancy, finite persistent address capacity bounds the total Jeffreys volume after finite-atlas globalization. If the pulled-back physical metric is exactly realized as \(h:=\iota^*h_{\mathrm{phys}}=(\ell^2/4)J^{\mathrm{geo}}\), sector-complete coverage gives \[\operatorname{Vol}_{h_{\mathrm{phys}}}(\Sigma_{\mathrm{phys}})\leq\frac{\ell^3}{8}\exp\left(\mathcal C_{\mathrm{addr}}^{\mathrm{nat}}\right)<\infty.\] A finite-mismatch theorem extends the result to a positive integrated metric-mismatch budget and an explicit residual-volume term. The construction retains every positive-volume represented or residual sector in a single physical ledger and applies to compact and noncompact finite-volume realizations without preselecting a topology. The theorem completes the missing comparison from operational information resource to global information volume and then to physical spatial volume. It is autonomous and supplies the realization criterion for information-projection cosmology: a parent branch is certified by the same-branch address family, finite common capacity, Fisher sufficiency, physical metric pullback, and residual closure. For every branch satisfying these explicit certificates, infinite spatial volume is mathematically excluded; geometry is a finite physical structure governed by the system's own operational capacity. This establishes the definitive global finite-volume theorem layer of the finite-universe series.