Cocycle Surface Holonomy and Horizon Completion: Phase Readout of Seams, Surface-Valued Cocycle Invariants, and Split Completion at Finite Area Budget

Assume hidden transport seams admit a phase-valued readout. This paper isolates the cohomological invariant sector selected by that hypothesis under bounded observability. We show that coherence forces a U(1)-valued 2-cocycle and that gauge ambiguity reduces to coboundary. The stabilized invariant datum is therefore a cohomology class in H^2(G;U(1)).

We then treat this class as surface-valued transport content. Given a branched triangulation of a closed oriented surface and a flat G-field, we define a surface holonomy as a product of local cocycle weights. The cocycle identity is exactly the local Pachner move relation, hence the holonomy is retriangulation invariant and depends only on the cohomology class and the induced map to BG. For genus g we identify the same phase as a lifted-commutator invariant in a corresponding central extension.

Finally we reintroduce loss and finite resources. On a finite lab 2-complex K, microstates are flat G-fields. A coarse phase readout and an area budget on 2-cycle queries induce a horizon indistinguishability relation. The resulting split quotients are the minimal observer-level refinements on which all bounded-area coarse phase queries become single-valued predictions. This furnishes a concrete bounded-observability regime for the universal completion principle; in finite regimes it also yields the exact horizon tower on which completion dynamics is defined once measures and rates are chosen.