13 Exact boundary of the article
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The formal results above sharpen the logical role of patching support, Hecke packets, and compatible systems. They do not replace the arithmetic theorems that supply the gates.
| Proved in this article | Not proved in this article |
|---|---|
| Local conditions combine through completed quotient rings | Representability of every desired local deformation condition |
| Full ordinary and derived support descends through the patching quotient | Near faithfulness or full support for every global deformation problem |
| An isomorphism \(R\cong{\mathbb T}\) transports support and Fitting data | An \(R\cong{\mathbb T}\) theorem outside published hypotheses |
| Finite Hecke algebras split into Artin packets | Positive automorphic occupancy of every residual or characteristic zero packet |
| Nonzero stable cores produce compatible inverse limit points | Universal residual companion existence |
| One automorphic member determines a fixed compatible system | Existence of a compatible system for every geometric representation |
| Augmentation and coinvariant recovery work at coefficient prime two | A universal coefficient two replacement for residual image, local geometry, and classicality hypotheses |
Remark 13.1
The strongest global conclusion is conditional automorphic realization inside a declared class for which the five gates have been verified. The article does not prove unrestricted global Langlands reciprocity.
Remark 13.2
A numerical certificate can check the finite algebra appearing in an example. It cannot verify a published automorphy lifting theorem, a classicality theorem, or a universal existence statement.