Abstract
On a supplied real Euclidean two-plane, admitting both rotations and reflections gives the transport group O(2) = U(1) ⋊ Z₂, not an untwisted circle. We organize the resulting graph, face, refinement, and readout structures in one assumption-controlled framework. On a connected finite graph, tree gauge reduces the edge data to a simultaneous conjugacy class of free generators. Its determinant class lies in H¹(Γ; Z₂), while the phase is a crossed cocycle for the associated sign action. Admitting faces introduces additional path comparisons: their determinant defects obey a relative boundary law, and a rooted tetrahedral word gives an exact non-Abelian Bianchi identity whose phase sum is sign-twisted. Shared-edge and common-root data permit exact boundary cancellation under the stated disk refinements. Explicit pentagonal and hexagonal certificates include all five and fourteen triangulations, respectively, and two mixed 1–3, 2–2, 3–1 routes. Mismatched-edge and omitted-root controls distinguish invalid fusion, while a preserved reverse-order null control shows why equality of evaluated matrices is weaker than equality of free boundary words. A locally flat annulus retains a nontrivial noncontractible reflection holonomy. Finally, conjugacy data, one prepared direction, a complete endpoint frame, and a separately supplied complex-linear process representation have different identification power. The component graph-gauge, cohomological, lattice-Bianchi, and triangulation structures have established antecedents. The proposed contribution is their explicit finite integration with typed root, edge-sharing, and access contracts. No arbitrary-mesh or continuum invariance, higher gauge theory, quantum-state selection, winding lift, probability law, or empirical realization follows from the construction.
Where it sits in the release
The finite-transport paper of the branch. The component graph-gauge, cohomological, lattice-Bianchi and triangulation structures have established antecedents; the contribution is their explicit finite integration under typed root, edge-sharing and access contracts.
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