Geometric Realization of Completed Source Relations: Descent, Orbit Spaces, Invariant Relations, and Variational Response in Shadow Theory

Summary

Identifies precisely when the abstract minimal completion of Paper 3 can be built from real geometry. Čech descent constructs global metrics, bundles, connections, and matter fields unique up to bundle isomorphism; gauge- and diffeomorphism-invariant relations such as holonomy, Wilson observables, characteristic numbers, and operator spectra descend to the orbit space of physical configurations and realize the completion on the restricted domain, with an exactly solved flat U(1) circle example where holonomy alone reconstructs the source though local curvature vanishes. Relation-dependent invariant actions then yield correctly normalized stress-energy, current, and matter responses, with Ward–Noether identities enforcing covariant conservation. The descent and cocycle conditions characterize the exact geometric realization domain and its failure modes.

Notes

Reading notes

Paper 3's completion is an abstract quotient. This paper identifies exactly when it possesses a genuine realization in differential geometry and classical field theory, including exact failure criteria.

The realization domain is built in three steps. First, geometric descent: compatible local metric, bundle, connection, and matter data, expressed as Čech cocycles plus overlap equations, glue to global fields on a four-manifold, unique up to bundle isomorphism, never up to a choice of chart or gauge. The corollary is the realizability boundary: an abstract tuple of relation values with no compatible descent data lies outside the geometric domain (a $U(1)$ bundle over $S^2$ with Chern number $\tfrac12$ fails at the cocycle stage). Second, the physical geometric source is the orbit space of admissible global configurations under the groupoid of bundle isomorphisms covering allowed diffeomorphisms; redundancy is removed before completion. Third, gauge- and diffeomorphism-invariant global relations such as holonomy conjugacy classes, Wilson observables, characteristic numbers, boundary data, and operator spectra descend to functions on that orbit space.

The flat $U(1)$ circle is solved exactly: holonomy classifies gauge orbits completely, so the realized completion is $E_{\mathrm{hol}} \cong U(1)$ and holonomy alone reconstructs the source even though the curvature readout distinguishes nothing. This is the mathematical content of the Aharonov–Bohm observation. A charged-scalar spectral relation instead yields the strictly coarser $E_{\mathrm{spec}} \cong [-1,1]$: Paper 2's target-relativity realized geometrically.

The final sections embed completed relations into variational dynamics: an invariant relation-dependent action yields correctly normalized stress-energy, gauge-current, and matter responses by functional variation. The responses are derived from the action, and Ward–Noether identities enforce covariant conservation of the total, functioning as selection rules on any candidate coupling.

Cite this paper

Rodgers, Jeremy. (2026). Geometric Realization of Completed Source Relations: Descent, Orbit Spaces, Invariant Relations, and Variational Response in Shadow Theory. https://doi.org/10.5281/zenodo.21370985