Conclusion

Conclusion

Canonical source completion acquires physical meaning only on a domain where its relation coordinates are jointly realized by compatible geometric data. Four results establish where that domain lies and what it supports.

The geometric descent theorem (Theorem 3.4) constructs the global metric, principal bundle, connection, and matter section from compatible local data, uniquely up to bundle isomorphism, with the global time-orientation condition treated explicitly (Remark 3.6); its corollary (Corollary 3.7) exhibits the realizability boundary, since relation values admitting no compatible descent data are simply outside the geometric realization domain. Passage to the orbit space of the bundle-automorphism groupoid (Definition 2.9) removes coordinate and gauge redundancy before completion, so that no gauge representative is taken for a physical state; the isotropy of a configuration is its stabilizer, which is in general a proper subgroup of the ambient automorphism group (Remark 2.7). The restricted geometric realization theorem (Theorem 5.3) then identifies the joint image of the readout and the declared invariant relations with the canonical minimal completion of Paper 3, relative to that family, and only on that domain, while the orbit reconstruction criterion (Proposition 5.5) reduces exact reconstruction to a separation question in invariant theory, answered case by case and never by fiat.

The flat-circle classification (Theorem 6.5 and Corollaries 6.7–6.9) shows in a fully solvable case that a global relation can completely parametrize a reduced source even when local curvature vanishes identically, and that a coarser declared relation leaves a strictly coarser completion behind. Finally, invariant relation-dependent actions (Theorem 7.8) convert completed structure into correctly normalized Einstein, Yang–Mills, and matter equations, with the associated covariant response fields derived rather than postulated, and the Ward–Noether identities (Proposition 8.3, Theorem 8.4, Proposition 8.5) enforce covariant conservation and supply a sharp necessary condition on any candidate realization.

This supplies the geometric bridge from abstract completion to observable Tier-1 structure. It does not assert that every abstract completion is geometrically realizable, that any finite family of local invariants separates Lorentzian geometries, or anything about the physical status of our universe. The dynamical questions of closure, retention, elimination, and memory belong to Paper 5.